LATTICE STATISTICS AND MATHEMATICAL PHYSICS
SERIES O N ADVANCES IN STATISTICAL MECHANICS Editor-in-Chief:
M. Rasetti
Published Vol. 1: Integrable Systems in Statistical Mechanics edited by G. M. D'Ariano, A. Montorsi & M. Rasetti Vol. 2:
Modern Methods in Equilibrium Statistical Mechanics by M. Rasetti
Vol. 3:
Progress in Statistical Mechanics edited by C. K. Hu
Vol. 4:
Thermodynamics of Complex Systems by L Sertorio
Vol. 5:
Potts Model and Related Problems in Statistical Mechanics by P. Martin
Vol. 6:
New Problems, Methods and Techniques in Quantum Field Theory and Statistical Mechanics edited by M. Rasetti
Vol. 7:
The Hubbard Model - Recent Results edited by M. Rasetti
Vol. 8:
Statistical Thermodynamics and Stochastic Theory of Nonlinear Systems Far From Equilibrium by W. Ebeling & L Schimansky-Geier
Vol. 9:
Disorder and Competition in Soluble Lattice Models by W. F. Wreszinski & S. R. A. Salinas
Vol. 10:
An Introduction to Stochastic Processes and Nonequilibrium Statistical Physics byH.S. Wio
Vol. 12:
Quantum Many-Body Systems in One Dimension by Zachary N. C. Ha
Vol. 13:
Exactly Soluble Models in Statistical Mechanics: Historical Perspectives and Current Status edited by C. King & F. Y. Wu
Vol. 14:
Statistical Physics on the Eve of the 21st Century: In Honour of J. B. McGuire on the Occasion of his 65th Birthday edited by M. T. Batchelor& L T. Wille
Vol. 15:
Lattice Statistics and Mathematical Physics: Festschrift Dedicated to Professor Fa-Yueh Wu on the Occasion of his 70th Birthday edited by J. H. H. Perk&M.-L Ge
Series on Advances in Statistical Mechanics - Volume 15
PROCEEDINGS JOINT
OF THE A P C T P - N A N K A I
SYMPOSIUM
LATTICE STATISTICS AND MATHEMATICAL PHYSICS Festschrift Dedicated to Professor Fa-Yueh Wu on the Occasion of His 70th Birthday
Tianjin, China
7 - 1 1 October 2001
edited by
Jacques H. H. Perk Oklahoma State University, USA
Mo-Lin Ge Nankai University, China
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International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) iii © World Scientific Publishing Company
PREFACE
This volume contains a collection of papers presented at the Nankai Symposium on "Lattice Statistics and Mathematical Physics," which was organized to honor the seventieth birthday of Professor Fa Yueh Wu ({ESS)- This conference took place at the Nankai Institute of Mathematics in Tianjin, China, hosted by its Vice Director Professor Mo-Lin Ge, October 7-11, 2001, co-organized with APCTP and Beijing Normal University. We are grateful to the support of the K. C. Wong Education Foundation, the APCTP and the NSF of China.
Jacques H. H. Perk Mo-Lin Ge May 2002
Fa-Yueh Wu (ffi&£)
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) v-xvi © World Scientific Publishing Company
FAYUEHWU*
(ffi$££r)
1. Brief Biography Fa-Yueh Wu (a.k.a. Fred Wu) was born January 1932. He moved with his parents and the Chinese government throughout the Sino-Japanese war and the civil war from 1938 to 1949. It may be noted that he graduated from Nankai Junior High School in Chungking, making him an "alumnus of Nankai." After graduating from high school in 1949, he eventually moved with his parents to Taiwan. There he entered the Chinese Naval College of Technology in 1949, obtaining a B.S. degree in Electrical Engineering in 1954, and receiving the commission as an Ensign in the navy. Wu was sent by the Chinese navy to the U.S. in 1955 to receive training at the Naval School of Electronics in San Francisco and the Instructors' School in San Diego, returning to Taiwan in 1956 to teach Electronics at the Naval Academy. He was a full-fledged expert on radar and sonar at that time, with a skill he has found useful recently in resoldering and fixing his broken remote car key. Fred Wu was (and probably still is) a good player of Chinese chess. He was a regional champion in Taiwan in 1951, and later the 1956 champion of all armed forces in Taiwan while a naval ensign. His favorite pastime in his graduate student years was to play chess "blind" with classmates while working on his homework. He has challenged the participants of the symposium to see if he is still as sharp as he used to be. But nobody took up the challenge. The very next year, in 1957, he entered the graduate school of the National Tsing Hua University in Taiwan, obtaining an M.S. degree in physics two years later. In 1959 he entered Washington University in St. Louis as a physics graduate student, where he studied under the late Professor Eugene Feenberg, working on many-body problems and obtaining his Ph.D. in physics in 1963. He taught for four years at Virginia Polytechnic Institute before coming to Northeastern University in 1967, where he is presently the Matthews University Distinguished Professor of Physics. ' P r e p a r e d by Jacques H.H. Perk, Department of Physics, Oklahoma State University, 145 P S , Stillwater, OK 74078-3072, USA, email:
[email protected]. Financial support by the conference organization, by the K. C. Wong Educational Foundation, and by NSF Grant No. PHY 01-00041 is gratefully acknowledged. vii
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Wu has accumulated a publication list a of over 200 papers and monographs. His earliest paper is in Chinese and published by him in 1955 while an Ensign in the Chinese navy. This paper 1 bears the title "On the discussions of 'free waveforms'." While Wu is known mostly for his publications in statistical mechanics, his works on many-body problems, especially those on liquid helium, have also been influential for many years, see e.g. Ref. 3, which was part of his Ph.D. thesis research. He has even published one experimental paper 5 with the title "Four slow neutron converters." Wu came to Northeastern to work with Elliott Lieb in 1967, and in 1968 they published a classic joint paper on the ground state of the Hubbard model. 11 This paper has become prominent in the theory of high-Tc superconductors. Anderson has attributed to it "predicting" the existence of quarks, in his Physics Today article on the Centennial of the discovery of electrons. Lieb and Wu also wrote a monograph on vertex models in 1970, which has become a principal reference in the field for decades.30 Since Wu came to the U.S. in 1959 as an ensign in the Chinese navy and was not decommissioned then, he was promoted in rank while a graduate student and a faculty member, eventually reaching the rank of Lieutenant in 1963. Therefore, much of his early work including the monograph with Lieb was done by a Lieutenant of the Chinese navy. Eventually, he could not be promoted further since for that he had to take an exam and the Navy was not sure whether he could pass it. He was later decommissioned from the rank of Navy Lieutenant in 1971. Thus the Chinese Navy saved a bunch of retirement benefits paid to retirees depending on the length of their service. Wu has worked on a wide-range of topics in many-body theory and statistical mechanics, including contributions in lattice statistics, graph theory, combinatorics, number theory, knot theory, and the interrelation between these topics. Wu's 1982 review on the Potts model is also well-known.88 This paper has been receiving over a hundred citations for many years ever since it was published.15 In 1992 Wu published another well-received review on knot theory. 153 Fred Wu has since been referred to as being "knotty" by Professor Lebowitz, which might be said to be a little "naughty" of Joel. 2. Some other selected publications Another classic is the paper on the Free Fermion Model. 16 This was later extended to its checkerboard version during one of Wu's many visits to Taiwan. 49 ' 51 Fred Wu was a close friend of the late Professor Piet Kasteleyn, who was co-advising my thesis work with Professor Hans Capel in Leiden at the time. Kasteleyn noted the a
T h i s list has been appended and a selection of this work has been cited in the following, reflecting the taste of the present editor. b I n 1982, the year the Potts review was published, it was the fifth most-cited paper among papers published in all of physics according to E. Garfield, [Current Comments 4 8 , 3 (1984)]. viii
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similarity of my first major paper 0 on the alternating XY-chain and the preprints of the above works. Both showed multiple phase transitions. Well-known is also the Baxter-Wu Model, i.e. the Ising model with three-spin interactions on a triangular lattice. 4 4 ' 4 8 Another classic paper, by Baxter, Kelland and Wu, concerns the graphical construction of the equivalence of the partition functions of the Potts model and a certain staggered six-vertex model. 56 Many people consider this construction easier than the algebraic method of Temperley and Lieb. Both methods are widely used these days. This paper is also at the basis of my first joint work with Fred Wu. 102 Here we generalized this equivalence to include the nonintersecting string (NIS) model of Stroganov and Schultz, alias the Close-Packed Loop Model. The six-vertex model is boundary-condition dependent. However, Brascamp, Kunz and Wu established for the first time that at sufficiently low temperatures or sufficiently high fields the six-vertex models with either periodic or free boundary conditions are equivalent. 42 Another remarkable result of Wu is that a very general staggered eight-vertex model in the Ising language (introduced in 1971 by Kadanoff and Wegner and by Wu 28 in two back-to-back papers), but with the special magnetic field mkB,T/2 of Lee and Yang added, is equivalent to Baxter's symmetric eight-vertex model and hence solvable. 104 The general eight-vertex model without this field is not known to be solvable. The dimer model on the honeycomb lattice was first solved by Kasteleyn. This has recently been generalized by Huang, Wu, Kunz and Kim to the case where the dimers have nearest-neighbor interaction. 172 This model relates to a degenerate case of the six-vertex model, requiring a special Bethe Ansatz analysis. The resulting phase diagram of this five-vertex model is quite complicated. This work has also been used in papers by Huang, Popkov and Wu on the three-dimensional dimer model. 1 7 5 ' 1 8 0 Its phase diagram is also quite complex. In 1999, Lu and Wu initiated work on dimer and Ising models on nonorientable surfaces, 1 9 1 ' 2 0 0 , 2 0 2 and generalized a reciprocity theorem in dimer combinatorics due to R. Stanley and J. Propp. 2 0 3 There is now much activity in this area, inspired by this work, as there is much interest in finite-size corrections and conformal field theories on more complicated surfaces. This is, of course, only a limited selection. A more precise understanding of the impact of Wu's work can be obtained by going over the following publication list and from the many papers in the volume. Therefore, I can speak on behalf of the other editor Professor Ge and the many participants of the symposium: Happy birthday and thank you, Professor Wu, for your many special insights and for being a friend of us all and not just a colleague.
C
J.H.H. Perk, H.W. Capel, M.J. Zuilhof, and Th.J. Siskens, Physica A 8 1 , 319-348 (1975).
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Publications of F.Y. Wu 1. F. Y. Wu, On the discussions of 'free waveforms', Naval Tech. 2, 9-11 (1955) [in Chinese]. 2. F. Y. Wu, Ground state of liquid He , in Proceedings of the Midwest Conference on Theoretical Physics, 182-196 (1961). 3. F. Y. Wu and E. Feenberg, Ground state of liquid helium (Mass 4), Phys. Rev. 122, 739-742 (1961). 4. F. Y. Wu and E. Feenberg, Theory of Fermi liquids, Phys. Rev. 128, 943-955 (1962). 5. F. Y. Wu, T. S. Kuo and K. H. Sun, Four slow neutron converters, Nucleonics 20, 61-62 (1962). 6. F. Y. Wu, The ground state of liquid He 4 , Prog. Theoret. Phys. (Kyoto) 28, 568-569 (1962). 7. F. Y. Wu, Cluster development in an iV-body problem, J. Math. Phys. 4, 1438-1443 (1963). 8. F. Y. Wu, H. T. Tan and E. Feenberg, Necessary conditions on radial distributions functions, J. Math. Phys. 8, 864-869 (1967). 9. F. Y. Wu, Exactly soluble model of ferroelectric phase transition in two dimensions, Phys. Rev. Lett, 18, 605-607 (1967). 10. F. Y. Wu, Remarks on the modified potassium dihydrogen phosphate (KDP) model, Phys. Rev. 168, 539-543 (1968). 11. E. Lieb and F. Y. Wu, Absence of Mott transition in an exact solution of the shortrange one-band model in one dimension, Phys. Rev. Lett. 20, 1445-1448 (1968). 12. C. Fan and F. Y. Wu, Ising model with next-neighbor interactions: Some exact results and an approximate solution, Phys. Rev. 179, 560-570 (1969). 13. F. Y. Wu, Critical behavior of two-dimensional hydrogen-bonded antiferroelectrics, Phys. Rev. Lett. 22, 1174-1176 (1969). 14. F. Y. Wu, Exact solution of a model of an antiferroelectric transition, Phys. Rev. 183, 604-607 (1969). 15. F. Y. Wu and M. K. Chien, Convolution approximation for the n-particle distribution function, J. Math. Phys. 11, 1912-1916 (1970). 16. C. Fan and F. Y. Wu, General Lattice Model of Phase Transitions, Phys. Rev. B 2, 723-733 (1970). 17. M. F. Chiang and F. Y. Wu, One-dimensional binary alloy with Ising interactions, Phys. Lett. A 31, 189-191 (1970). 18. F. Y. Wu, Critical behavior of hydrogen-bonded ferroelectrics, Phys. Rev. Lett. 24, 1476-1478 (1970). 19. F. Y. Wu, C. W. Woo and H. W. Lai, Communications on a microscopic theory of helium submonolayers, J. Low Temp. Phys. 3, 331-333 (1970). 20. F. Y. Wu, H. W. Lai and C. W. Woo, Theory of helium submonolayers: Formulation of the problem and solution in the zero coverage limit, J. Low Temp. Phys. 3, 463-490 (1970). 21. F. Y. Wu, Density correlations in a many particle system, Phys. Lett. A 34, 446-447 (1971). 22. F. Y. Wu and D. M. Kaplan, On the eigenvalues of orbital angular momentum, Ch. J. Phys. 9, 31-40 (1971). 23. F. Y. Wu, Modified potassium dihydrogen phosphate model in a staggered field, Phys. Rev. B 3, 3895-3900 (1971). 24. F. Y. Wu, H. T. Tan and C. W. Woo, Sum rules for a binary solution and effective interactions between He quasiparticles in superfluid He 4 , J. Low Temp. Phys. 5, 261-265 (1971). x
F.-Y. Wu ix 25. F. Y. Wu, H. W. Lai and C. W. Woo, Theory of helium submonolayers II: Bandwidths and correlation effects, J. Low Temp. Phys. 5, 499-518 (1971). 26. F. Y. Wu and C. W. Woo, Physically adsorbed monolayers, Ch. J. Phys. 9, 68-91 (1971). 27. F. Y. Wu, Multiple density correlations in a many particle system, J. Math. Phys. 12, 1923-1929 (1971). 28. F. Y. Wu, Ising model with four-spin interactions, Phys. Rev. B 4, 2312-2314 (1971). 29. F. Y. Wu, K. S. Chang and S. Y. Wang, Circle theorem for the ice-rule ferroelectric models, Phys. Rev. A 4, 2324-2327 (1971). 30. E. H. Lieb and F. Y. Wu, Two-dimensional ferroelectric models, in Phase Transitions and Critical Phenomena, Vol. 1, Eds. C. Domb and M. S. Green (Academic Press, London, 1972) pp. 331-490. 31. F. Y. Wu, Exact results on a general lattice statistical model, Solid State Commun. 10, 115-117 (1972). 32. F. Y. Wu, Solution of an Ising model with two- and four-spin interactions, Phys. Lett. A 38, 77-78 (1972). 33. F. Y. Wu, Phase transition in a sixteen-vertex lattice model, Phys. Rev. B 6, 18101813 (1972). 34. G. Keiser and F. Y. Wu, Electron gas at metallic densities, Phys. Rev. A 6, 2369-1377 (1972). 35. F. Y. Wu, Matrix elements of a many-boson Hamiltonian in a representation of correlated basis functions, J. Low Temp. Phys. 9, 177-187 (1972). 36. F. Y. Wu, Statistics of parallel dimers on an m x n lattice, Phys. Lett. A 4 3 , 21-22 (1973). 37. H. J. Brascamp, H. Kunz and F . Y. Wu, Critical exponents of the modified F model, J. Phys. C 6 , L164-L166 (1973). 38. K. S. Chang, S. Y. Wang and F . Y. Wu, Statistical mechanics of finite ice-rule ferroelectric models, Ch. J. Phys. 1 1 , 41-48 (1973). 39. F. Y. Wu, Phase diagram of a decorated Ising system, Phys. Rev. B 8, 4219-4222 (1973). 40. F. Y. Wu, Modified KDP model on the Kagome lattice, Prog. Theor. Phys. (Kyoto) 49, 2156-2157 (1973). 41. G. Keiser and F. Y. Wu, Correlation energy of an electron gas in the quantum strong-field limit, Phys. Rev. A 8, 2094-2098 (1973). 42. H. J. Brascamp, H. Kunz and F. Y. Wu, Some rigorous results for the vertex model in statistical mechanics, J. Math. Phys. 14, 1927-1932 (1973). 43. F. Y. Wu, Phase transition in an Ising model with many-spin interactions, Phys. Lett. A 46, 7-8 (1973). 44. R. J. Baxter and F. Y. Wu, Exact solution of an Ising model with three-spin interactions on a triangular lattice, Phys. Rev. Lett. 3 1 , 1294-1297 (1973). 45. F. Y. Wu, Phase transition in a vertex model in three dimensions, Phys. Rev. Lett. 32, 460-463 (1974). 46. F. Y. Wu and K. Y. Lin, Two phase transitions in the Ashkin-Teller model, J. Phys. C 7 , L181-L184 (1974). 47. F. Y. Wu, Eight-vertex model on the honeycomb lattice, J. Math. Phys. 6, 687-691 (1974). 48. R. J. Baxter and F. Y. Wu, Ising model on a triangular lattice with three-spin interactions: I. The eigenvalue equations, Aust. J. Phys. 27, 357-367 (1974). 49. F. Y. Wu and K. Y. Lin, Staggered ice-rule model—The Pfaffian solution, Phys. Rev. B 12, 419-428 (1975).
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F.-Y. Wu 50. F. Y. Wu, Spontaneous magnetization of the three-spin Ising model on the Union Jack lattice, J. Phys. C 8, 2262-2266 (1975). 51. C. S. Hsue, K. Y. Lin and F. Y. Wu, Staggered eight-vertex model, Phys. Rev. B 12, 429-437 (1975). 52. J. E. Sacco and F. Y. Wu, Thirty-two vertex model on a triangular lattice, J. Phys. A 8, 1780-1787 (1975). 53. Y. K. Wang and F. Y. Wu, Multi-component spin model on the Bethe lattice, IUPAP Conference on Statistical Physics (Akademiai Kiado, 1975), p. 142. 54. M. L. C. Leung, B. Y. Tong and F. Y. Wu, Thermal denaturation and renaturation of DNA molecules, Phys. Lett. A 54, 361-362 (1975). 55. Y. K. Wang and F. Y. Wu, Multi-component spin model on the Cayley tree, J. Phys. A 9, 593-604 (1976). 56. R. J. Baxter, S.B. Kelland and F. Y. Wu, Equivalence of the Potts model or Whitney polynomial with the ice-type model: A new derivation, J. Phys. A 9, 397-406 (1976). 57. F. Y. Wu and Y. K. Wang, Duality transformation in a many component spin model, J. Math. Phys. 17, 439-440 (1976). 58. F. Y. Wu, Two phase transitions in triplet Ising models, J. Phys. C 10, L23-L27 (1977). 59. F. Y. Wu, Ashkin-Teller model as a vertex problem, J. Math. Phys. 18, 611-613 (1977). 60. K. G. Chen, H. H. Chen, C. S. Hsue and F. Y. Wu, Planar classical Heisenberg model with biquadratic interactions, Physica A 87, 629-632 (1977). 61. F. Y. Wu, Number of spanning trees on a lattice, J. Phys. A 10, L113-L115 (1977). 62. H. Kunz and F. Y. Wu, Site percolation as a Potts model, J. Phys. C 11, L1-L4 (1978). 63. F. Y. Wu, Percolation and the Potts model, J. Stat. Phys. 18, 115-123 (1978). 64. F. Y. Wu, Some exact results for lattice models in two dimensions, Ann. Israel Phys. Soc. 2, 370-376 (1978). 65. V. T. Rajan, C.-W. Woo and F. Y. Wu, Multiple density correlation for inhomogeneous systems, J. Math. Phys. 19, 892-897 (1978). 66. F. Y. Wu, Absence of phase transitions in tree-like percolation in two dimensions, Phys. Rev. B 18, 516-517 (1978). 67. F. Y. Wu, Phase diagram of a spin-one Ising system, Ch. J. Phys. 16, 153-156 (1978). 68. F. Y. Wu, On the Temperley-Nagle identity for graph embeddings, J. Phys. A 11, L243 (1978). 69. A. Hinterman, H. Kunz and F. Y. Wu, Exact results for the Potts model in two dimensions, J. Stat. Phys. 19, 623-632 (1978). 70. F.Y. Wu, Graph theory in statistical physics, in Studies in Foundations and Combinatorics, Adv. in Math: Supple. V.I, Ed. G.-C. Rota (Academic Press, New York 1978), pp. 151-166. 71. K. Y. Lin and F. Y. Wu, Phase diagram of the antiferromagnetic triangular Ising model with anisotropic interactions, Z. Phys. B 33, 181-185 (1979). 72. F. Y. Wu, Phase diagram of a five-state spin model, J. Phys. C12, L317-L320 (1979). 73. F. Y. Wu, Critical point of planar Potts models, J. Phys. C 12, 645-649 (1979). 74. X. Sun and F. Y. Wu, Critical polarization of the modified F model, J. Phys. C 12, L637-L641 (1979). 75. F. Y. Wu and K. Y. Lin, On the triangular Potts model with two- and three-site interactions, J. Phys. A14, 629-636 (1980). 76. F. Y. Wu, Exact results for a dilute Potts model, J. Stat. Phys. 23, 773-782 (1980).
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F.-Y. Wu xi 77. F. Y. Wu, On the equivalence of the Ising model with a vertex model, J. Phys. A 13, L303-L305 (1980). 78. R. K. P. Zia and F. Y. Wu, Critical point of the triangular Potts model with twoand three-site interactions, J. Phys. A 14, 721-727 (1981). 79. X. Sun and F. Y. Wu, The critical isotherm of the modified F model, Physica A 106, 292-300 (1981). 80. F. Y. Wu, Dilute Potts model, duality and site-bond percolation, J. Phys. A 14, L39-L44 (1981). 81. W. Kinzel, W. Selke and F. Y. Wu, A Potts model with infinitely degenerate ground state, J. Phys. A 14, L399-L404 (1981). 82. S. Sarbach and F. Y. Wu, Exact results on the random Potts model, Z. Phys. B 44, 309-316 (1981). 83. F. Y. Wu and Z. R. Yang, Critical phenomena and phase transitions I: Introduction, Prog, in Physics (in Chinese) 1, 100-124 (1981). 84. F. Y. Wu and Z. R. Yang, Critical phenomena and phase transitions II: The Ising Model, Prog, in Physics (in Chinese) 1, 314-364 (1981). 85. F. Y. Wu and Z. R. Yang, Critical phenomena and phase transitions III: The vertex model, Prog, in Physics (in Chinese) 1, 487-510 (1981). 86. F. Y. Wu and Z. R. Yang, Critical phenomena and phase transitions IV: Percolation, Prog, in Physics (in Chinese) 1, 511-525 (1981). 87. F. Y. Wu and Z. R. Yang, Critical phenomena and phase transitions V: Model systems, Prog, in Physics (in Chinese) 1, 525-541 (1981). 88. F. Y. Wu, The Potts Model, Rev. Mod. Physics 54, 235-268 (1982). 89. F. Y. Wu, and H. E. Stanley, Domany-Kinzel model of directed percolation: Formulation as a random-walk problem and some exact results, Phys. Rev. Lett. 48, 775-777 (1982). 90. I. G. Enting and F. Y. Wu, Triangular lattice Potts model, J. Stat. Phys. 28, 351-378 (1982). 91. F. Y. Wu, Random graphs and network communication, J. Phys. A 15, L395-L398 (1982). 92. F. Y. Wu, An infinite-range bond percolation, J. Appl. Phys. 5 3 , 7977 (1982). 93. F. Y. Wu and H. E. Stanley, Universality in Potts models with two- and three-site interactions, Phys. Rev. B 26, 6326-6329 (1982). 94. F. Y. Wu and Z. R. Yang, The Slater model of K{B.\-xDx)iPOi in two dimensions, J. Phys. C16, L125-L129 (1983). 95. F. Y. Wu and H. E. Stanley, Polychromatic Potts model: A new lattice-statistical problem and some exact results, J. Phys. A 16, L751-755 (1983). 96. J. R. Banavar and F. Y. Wu, Antiferromagnetic Potts model with competing interactions, Phys. Rev. B 29, 1511-1513 (1984). 97. F. Y. Wu, Potts model of ferromagnetism, J. Appl. Phys. 55, 2421-2425 (1984). 98. D. H. Lee, R. G. Caflish, J. D. Joannopoulos, and F. Y. Wu, Antiferromagnetic classical XY-model: A mean-field analysis, Phys. Rev. B 29, 2680-2684 (1984). 99. F. Y. Wu, Exact solution of a triangular Ising model in a nonzero magnetic field, J. Stat. Phys. 40, 613-620 (1985). 100. N. C. Chao and F. Y. Wu, Disorder solution of a general checkerboard Ising model in a field and validity of the decimation approach, J. Phys. A 18, L603-607 (1985). 101. Z. R. Yang and F. Y. Wu, Exact solution of a dilute-bond Potts model on the decorated square lattice, Acta Physica Sinica (in Chinese), 34 484-492 (1985). 102. J. H. H. Perk and F. Y. Wu, Nonintersecting string model and graphical approach: Equivalence with a Potts model, J. Stat. Phys. 42, 727-742 (1986). xiii
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103. J. H. H. Perk and F. Y. Wu, Graphical approach to the nonintersecting string model: Star-triangle equation, inversion relation and exact solution, Physica A 138, 100-124 (1986). 104. F. Y. Wu, Two-dimensional Ising model with crossing and four-spin interactions and a magnetic field iirkT/2, J. Stat. Phys. 44, 455-463 (1986). 105. F. Y. Wu, On the Horiguchi's Solution of the Blume-Emery-Griffiths Model, Phys. Lett A 116, 245-247 (1986). 106. F. Y. Wu, Thermodynamics of particle in a box, Ann. Report Inst. Phys. Academia Sinica 16, 25-30 (1986). 107. H. K. Sim, R. Tao, and F. Y. Wu, Ground-state energy of charged quantum liquids in two dimensions, Phys. Rev. B 34, 7123-7128 (1986). 108. W. Selke and F. Y. Wu, Potts models with competing interactions, J. Phys. A 20, 703-711 (1987). 109. Y. Chow and F. Y. Wu, Residual entropy and validity of the third law of thermodynamics in discrete spin systems, Phys. Rev. B 36, 285-288 (1987). 110. R. Tao and F. Y. Wu, The Vicious Neighbor Problem, J. Phys. A 20, L299-L306 (1987). 111. G. O. Zimmerman, A. K. Ibrahim, and F. Y. Wu, The effect of defects on a twodimensional dipolar system on a honeycomb lattice, J. Appl. Phys. 61, 4416-4418 (1987). 112. F. Y. Wu, Book Review. Statistical mechanics of periodic frustrated Ising systems (R. Liebmann), J. Stat. Phys. 48, 953 (1987). 113. F. Y. Wu and K. Y. Lin, Ising model on the Union Jack lattice as a free-fermion model, J. Phys. A 20, 5737-5740 (1987). 114. G. O. Zimmerman, A. K. Ibrahim, and F. Y. Wu, A planar classical dipolar system on a honeycomb lattice, Phys. Rev. B 37, 2059-2065 (1988). 115. X. N. Wu and F. Y. Wu, The Blume-Emery-Griffiths Model on the honeycomb lattice, J. Stat. Phys. 50, 41-55 (1988). 116. F. Y. Wu, Potts model and graph theory, J. Stat. Phys. 52, 99-112 (1988). 117. K. Y. Lin and F. Y. Wu, Magnetization of the Ising model on the generalized checkerboard lattice, J. Stat. Phys. 52, 669-677 (1988). 118. K. Y. Lin and F. Y. Wu, Ising models in the magnetic field iir/2, Int. J. Mod. Phys. B 2, 471-481 (1988). 119. F. Y. Wu and K. Y. Lin, Spin model exhibiting multiple disorder points, Phys. Lett. A 130, 335-337 (1988). 120. H. Kunz and F. Y. Wu, Exact results for an 0(n) model in two dimensions, J. Phys. A 21, L1141-L1144 (1988). 121. F. Y. Wu, Potts model and graph theory, in Progress in Statistical Mechanics, Ed. C. K. Hu (World Scientific, Singapore 1988). 122. K. Y. Lin and F. Y. Wu, Three-spin correlation of the free-fermion model, J. Phys. A 22 1121-1130 (1989). 123. X. N. Wu and F. Y. Wu, Duality properties of a general vertex model, J. Phys. A 22, L55-L60 (1989). 124. F. Y. Wu, X. N. Wu and H. W. J. Blote, Critical frontier of the antiferromagnetic Ising model in a nonzero magnetic field: The honeycomb lattice, Phys. Rev. Lett. 62, 2273-2276 (1989). 125. F. Y. Wu and X. N. Wu, Exact coexistence-curve diameter of a lattice gas with pair and triplet interactions, Phys. Rev. Lett. 63, 465 (1989). 126. X. N. Wu and F. Y. Wu, Exact results for lattice models with pair and triplet interactions, J. Phys. A 22, L1031-L1035 (1989). xiv
F.-Y. Wu xiii 127. J. H. Barry and F. Y. Wu, Exact solutions for a four-spin-interaction Ising model on the three-dimensional pyrochlore lattice, Int. J. Mod. Phys. B 8, 1247-1275 (1989). 128. F. Lee, H. H. Chen and F. Y. Wu, Exact duality-decimation transformation and real-space renormalization for the Ising model on a square lattice, Phys. Rev. B 40, 4871-4876 (1989). 129. F. Y. Wu, Eight-vertex model and Ising model in a nonzero magnetic field on the honeycomb lattice, J. Phys. A 23, 375-378 (1990). 130. J. H. H. Perk, F. Y. Wu and X. N. Wu, Algebraic invariants of the symmetric gauge transformation, J. Phys. A 23, L131-L135 (1990). 131. H. W. J. Blote, F. Y. Wu and X. N. Wu, Critical point of the honeycomb antiferromagnetic Ising model in a nonzero magnetic field: Finite-size analysis, Int. J. Mod. Phys. B 4, 619-630 (1990). 132. A. C. N. de Magalhaes, J. W. Essam and F. Y. Wu, Duality relation for Potts multispin correlation functions, J. Phys. A 23, 2651-2669 (1990). 133. K. Y. Lin and F. Y. Wu, General eight-vertex model on the honeycomb lattice, Mod. Phys. Lett. B 4, 311-316 (1990). 134. X. N. Wu and F. Y. Wu, Critical line of the square-lattice Ising model, Phys. Lett. A 144, 123-126 (1990). 135. F. Y. Wu, Comment on 'Rigorous Solution of a Two-Dimensional Blume-EmeryGriffiths Model', Phys. Rev. Lett. 64, 700 (1990). 136. F. Y. Wu, Yang-Lee edge in the high-temperature limit, Ch. J. Phys. 28, 335-338 (1990). 137. K. Y. Lin and F. Y. Wu, Unidirectional-convex polygons on the honeycomb lattice, J. Phys. A 2 3 , 5003-5010 (1990). 138. F. Y. Wu, Ising model, in Encyclopedia of Physics, Eds. R. G. Lerner and G. L. Trigg (Addison-Wesley, New York 1990). 139. J. L. Ting, S. C. Lin and F. Y. Wu, Exact analysis of a lattice gas on the 3—12 lattice with two- and three-site interactions, J. Stat. Phys. 62, 35-43 (1991). 140. D. L. Strout, D. A. Huckaby and F. Y. Wu, An exactly solvable model ternary solution with three-body interactions, Physica A 173, 60-71 (1991). 141. F. Y. Wu, Rigorous results on the Ising model on the triangular lattice with twoand three-spin interactions, Phys. Lett. A 153, 73-75 (1991). 142. J. C. A. d'Auriac, J.-M. Maillard, and F. Y. Wu, Three-state chiral Potts models in two dimensions: Integrability and symmetry, Physica A 177, 114-122 (1991). 143. J. C. A. d'Auriac, J.-M. Maillard, and F. Y. Wu, Three-state chiral Potts model on the triangular lattice: A Monte Carlo study, Physica A 179, 496-506 (1991). 144. K. Y. Lin and F. Y. Wu, Rigorous results on the anisotropic Ising model, Int. J. Mod. Phys. B 5, 2125-2132 (1991). 145. L. H. Gwa and F. Y. Wu, The 0(3) gauge transformations and three-state vertex models, J. Phys. A 24, L503-L507 (1991). 146. J.-M. Maillard, F. Y. Wu and C. K. Hu, Thermal transmissivity in discrete spin systems: Formulation and applications, J. Phys. A 25, 2521-2531, (1992). 147. L. H. Gwa and F. Y. Wu, Critical surface of the Blume-Emery-Griffiths model on the honeycomb lattice, Phys. Rev. B 43, 13755-13777 (1991). 148. C. K. Lee, A. S. T. Chiang and F. Y. Wu, Lattice model for the adsorption of benzene in silicalite I, J. Am. Int. Chem. Eng. 38, 128-135 (1992). 149. F. Y. Wu, Knot theory and statistical mechanics, AIP Conf. Proc, No. 248, ed. C. K. Hu, 3-11 (1992). 150. F. Y. Wu, Jones polynomial as a Potts model partition function, J. Knot Theory and Its Ramifications 1, 47-57 (1992)
xiv F.-Y. Wu 151. F. Y. Wu, Transformation properties of a spin-1 system in statistical physics, Ch. J. Phys. 30, 157-163 (1992). 152. F. Y. Wu, Jones polynomial and the Potts model, He.lv. Phys. Acta 65, 469-470 (1992). 153. F. Y. Wu, Knot theory and statistical mechanics, Rev. Mod. Phys. 64, 1099-1131 (1992). 154. J.-M. Maillard, G. RoUet and F. Y. Wu, An exact critical frontier for the Potts model on the 3 - 1 2 lattice, J. Phys. A 26, L495-L499 (1993). 155. F. Y. Wu, The Yang-Baxter equation in knot theory, Int. J. Mod. Phys. B 7, 37373750 (1993). 156. F. Y. Wu and H. Y. Huang, Exact solution of a vertex model in d dimensions, Lett. Math. Phys. 29, 205-213 (1993). 157. F. Y. Wu, Knot theory and statistical mechanics: A physicist's perspective, in Braid Group, Knot Theory and Statistical Mechanics, Eds. M. L. Ge and C. N. Yang (World Scientific, Singapore 1993). 158. J. C. A. d'Auriac, J.-M. Maillard, G. RoUet, and F. Y. Wu, Zeroes of the partition function of the triangular Potts model: Conjectured distribution, Physica A 206, 441-453 (1994). 159. H. Y. Huang, and F. Y. Wu, Exact solution of a model of type-II superconductors, Physica A 205, 31-40 (1994). 160. J.-M. Maillard, G. RoUet, and F. Y. Wu, Inversion relations and symmetry groups for Potts models on the triangular lattice, J. Phys. A 27, 3373-3379 (1994). 161. F. Y. Wu, P. Pant, and C. King, A new knot invariant from the chiral Potts model, Phys. Rev. Lett. 72, 3937-9340 (1994). 162. C. K. Hu, J. A. Chen, and F. Y. Wu, Histogram Monte Carlo renormaUzation group approach to the Potts model on the Kagome lattice, Mod. Phys. Lett. B 8, 455-459 (1994). 163. F. Y. Wu, P. Pant and C. King, Knot invariant and the chiral Potts model, J. Stat. Phys. 78, 1253-1276 (1995). 164. F. Y. Wu and H. Y. Huang, Soluble free-fermion model in d dimensions, Phys. Rev. £ 5 1 , 889-895 (1995). 165. H. Meyer, J. C. A. d'Auriac, J.-M. Maillard, G. RoUet, and F. Y. Wu, Tricritical behavior of the three-state triangular Potts model, Phys. Lett. A 201, 252-256 (1995). 166. F. Y. Wu, Statistical mechanics and the knot theory, AIP Conf. Proc. No. 342, ed. A. Zepeda, 750-756 (1995). 167. G. Rollet and F. Y. Wu, Gauge transformation and vertex models: A polynomial formulation, Int. J. Mod. Phys. B 9, 3209-3217 (1995). 168. C. N. Chen, C. K. Hu, and F. Y. Wu, Partition function zeroes of the square lattice Potts model, Phys. Rev. Lett. 76 169-172 (1996). 169. C. K. Hu, C. N. Chen, and F. Y. Wu, Histogram Monte Carlo renormaUzation group: Applications to the site percolation, J. Stat. Phys. 82, 1199-1206 (1996). 170. F. Y. Wu, G. Rollet, H. Y. Huang, J.-M. Maillard, C. K. Hu and C. N. Chen, Directed compact lattice animals, restricted partitions of numbers, and the infinite-state Potts model, Phys. Rev. Lett. 76, 173-176 (1996). 171. A. S. T. Chiang, C. K. Lee, and F. Y. Wu, Theory of adsorbed solutions: analysis of one-dimensional systems, J. Am. Chem. Eng. 42, 2155-2161 (1996). 172. H. Y. Huang, F. Y. Wu, H. Kunz, and D. Kim, Interacting dimers on the honeycomb lattice: An exact solution of the five-vertex model, Physica A 228, 1-32 (1996). 173. F. Y. Wu, Soluble free-fermion models in d dimensions, in Statistical Models, YangBaxter Equation and Related Topics, and Symmetry, Statistical Mechanical Models
XVI
F.-Y. Wu
174. 175. 176. 177. 178. 179. 180.
181. 182. 183. 184. 185.
186. 187. 188. 189. 190. 191. 192. 193.
194.
195.
xv
and Applications, Eds. M. L. Ge and F. Y. Wu, (World Scientific, Singapore 1996), pp. 334-339. C. King and F. Y. Wu, Star-star relation for the Potts model, Int. J. Mod. Phys. B 1 1 , 51-56 (1997). H. Y. Huang, V. Popkov, and F. Y. Wu, Exact solution of a three-dimensional dimer system, Phys. Rev. Lett. 78, 409-413 (1997). H. Y. Huang and F. Y. Wu, The infinite-state Potts model and solid partitions of an integer, Int. J. Mod. Phys. B 11, 121-126 (1997). P. Pant, F. Y. Wu, and J. H. Barry, Exact results for a model ternary polymer mixture, Int. J. Mod. Phys. B 11, 169-174 (1997). J. H. Barry, P. Pant, and F. Y. Wu, A lattice-statistical model for ternary polymer mixtures: Exact phase diagrams, Physica A 238, 149-162 (1997). F. Y. Wu, Duality relations for Potts correlation functions, Phys. Lett. A 228, 43-47 (1997). V. Popkov, D. Kim, H. Y. Huang, and F. Y. Wu, Lattice statistics in three dimensions: Exact solution of layered dimer and layered domain wall models, Phys. Rev. E 56, 3999-4008 (1997). P. Pant and F. Y. Wu, Link invariant of the Izergin-Korepin Model, J. Phys. A 30, 7775-7782 (1997). F. Y. Wu, The infinite-state Potts model and restricted multidimensional partitions of an integer, Mathl. and comp. Modeling 26, 269-274 (1997). F. Y. Wu and H. Y. Huang, New sum rule identities and duality relation for the Potts n-point correlation function, Phys. Rev. Lett. 79, 4954-4957 (1997). F. Y. Wu and W. T. Lu, Correlation duality relations of the (iVa,./^) model, Ch. J. Phys. 36, (1998). F. Y. Wu, The exact solution of a class of three-dimensional lattice statistical model, in Proceedings 1th Asia-Pacific Physics Conference, Ed. H. Chen (Science Press, Beijing 1998). W. T. Lu and F. Y. Wu, On the duality relation for correlation functions of the Potts model, J. Phys. A 31, 2823-2836 (1998). W. T. Lu and F. Y. Wu, Partition function zeroes of a self-dual Ising model, Physica A 258, 157-170 (1998). C. A. Chen, D. K. Hu and F. Y. Wu, Critical point of the kagome Potts model, J. Phys. A 3 1 , 7855-7864 (1998). F. Y. Wu and H. Kunz, Restricted random walks on a graph, Ann. combinatorics 3, 475-481 (1999). F. Y. Wu, C. King, and W. T. Lu, On the rooted Tutte polynomial, Ann. Inst. Fourier, Grenoble 49, 101-112 (1999). W. T. Lu and F. Y. Wu, Dimer statistics on a Mobius strip and the Klein bottle, Phys. Letter. A 259, 108-114 (1999). F. Y. Wu, Exactly soluble models in statistical mechanics, Notices of AMS, 13541354 (1999). F. Y. Wu, Two soluble lattice models in three dimensions, in Statistical Physics at the Eve of the 21th Century, Eds. M. T. Batchelor and T. Wille (World Scientific, Singapore 1999), pp. 336-350. F. Y. Wu, Self-dual polynomials in statistical physics and number theory, in YangBaxter systems, nonlinear models and their applications, Eds. B. K. Chung, Q-Han Park, and C. Rim (World Scientific, Singapore 2000), pp. 77-82. W. J. Tzeng and F. Y. Wu, Spanning trees on hypercubic lattices and non-orient able surfaces, Appl. Math. Lett. 13, 19-25 (2000). xvii
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196. N. Sh. Izamailian, C. K. Hu, and F. Y. Wu, The 6-vertex model of hydrogen-bonded crystals with bond defects, J. Phys. A 33, 2185-2193 (2000). 197. R. Shrock and F. Y. Wu, Spanning trees on graphs and lattices in d dimensions, J. Phys. A 33, 3881-3902 (2000). 198. W. Guo, H. W. J. Blote, and F. Y. Wu, Phase transitions in the n > 2 honeycomb 0(n) model, Phys. Rev. Lett. 85, 3874-3811 (2000). 199. W. T. Lu and F. Y. Wu, Density of the Fisher zeroes for the Ising model, J. Stat. Phys. 102, 953-970 (2000). 200. W. T. Lu and F. Y. Wu, Ising model on non-orientable surfaces, Phys. Rev. E 63 026107 (2001). 201. F. Y. Wu and J. Wang, Zeroes of the Jones polynomial, Physica A 296, 483-494 (2001). 202. W. T. Lu and F. Y. Wu, Close-packed dimers on non-orientable surfaces, Phys. Lett A 293, 235-246 (2002). 203. C. King and F. Y. Wu, New correlation relations for the planar Potts model, J. Stat. Phys., to appear (2002). 204. W. J. Tzeng and F. Y. Wu, Dimers on a simple-quartic net with a vacancy, J. Stat. Phys., to appear (2002). 205. F. Y. Wu, Dimers and spanning trees: Some recent results, Int. J. Mod. Phys. B , this issue (2002).
XVlll
CONTENTS
Preface
v
Introduction: Fa Yueh Wu
vii
Star-Triangle Equations and Identities in Hypergeometric Series H. Au-Yang and J. H. H. Perk
1
Happer's Curious Degeneracies and Yangian C.-M. Bai, M.-L. Ge and K. Xue
15
Fractional Statistics in Some Exactly Solvable Models With PT Invariant Interactions B. Basu-Mallick
23
The Rotor Model and Combinatorics M. T. Batchelor, J. de Gier and B. Nienhuis
31
Phase Transitions in the Two-Dimensional 0(3) Model H. W. J. Blote and W. Guo
39
The 8V CSOS Model and the sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity T. Deguchi
47
The Chern-Simons Invariant in the Berry Phase of A Two by Two Hamiltonian D.-H. Lee
55
Mutually Local Fields from Form Factors A. Fring
63
Deformation Quantization: Is C\ Necessarily Skew? C. Fronsdal
73
Generalized Ginsparg-Wilson Algebra and Index Theorem on the Lattice K. Fujikawa
79
Spectral Flow of Hermitian Wilson-Dirac Operator and the Index Theorem in Abelian Gauge Theories on Finite Lattices T. Fujiwara
91
Dimers and Spanning Trees: Some Recent Results F. Y. Wu
99
xix
Hyperbolic Structure Arising from a Knot Invariant II: Completeness K. Hikami
Ill
Exotic Galilean Symmetry and the Hall Effect C. Duval and P. A. Horvathy
119
The Three-State Chiral Clock Model B.-Q. Jin, H. Au-Yang and J. H. H. Perk
127
Stochastic Description of Agglomeration and Growth Processes in Glasses R. Kerner
135
Fusion Construction of the Vertex Operators in Higher Level Representation of the Elliptic Quantum Group H. Konno
143
Quantum Dynamics and Random Matrix Theory H. Kunz
151
Integrable Coupling in a Model for Josephson Tunneling Between Non-Identical BCS Systems J. Links and K. E. Hibberd
157
Finite Density Algorithm in Lattice QCD — A Canonical Ensemble Approach K.-F. Liu
165
Short-Time Behaviors of Long-Ranged Interactions H. Fang, C. S. He, Z. B. Li, S. P. Seto, Y. Chen and M. Y. Wu
181
Generalized Supersymmetries and Composite Structure in J. Lukierski
187
M-Theory
Gaussian Orthogonal Ensemble for the Level Spacing Statistics of the Quantum Four-State Chiral Potts Model J.-Ch. Angles d'Auriac, S. Dommange, J.-M. Maillard and C. M. Viallet . 195 Comments on the Deformed WM Algebra S. Odake
203
The Peierls Substitution and the Vanishing Magnetic Field Limit S. Ouvry
213
Applications of a Hard-Core Bose-Hubbard Model to Well-Deformed Nuclei Y. Chen, F. Pan, G. S. Stoitcheva and J. P. Draayer
219
XX
Elliptic Algebra and Integrable Models for Solitons on Noncommutative Torus T B.-Y. Hou and D.-T. Peng
227
New Results for Susceptibilities in Planar Ising Models H. Au-Yang and J. H. H. Perk
237
Algebraic Geometry and Hofstadter Type Model S.-S. Lin and S.-S. Roan
245
Limitations on Quantum Control A. I. Solomon and S. G. Schirmer
255
Rigorous Results on the Strongly Correlated Electron Systems by the Spin-Reflection-Positivity Method G.-S. Tian
261
An Algebraic Approach to the Eigenstates of the Calogero Model H. Ujino
269
Onsager's Algebra and Partially Orthogonal Polynomials G. Von Gehlen
277
Thermodynamics of Two Component Bosons in One Dimension S.-J. Gu, Y.-Q. Li, Z.-J. Ying and X.-A. Zhao
285
R-Matrices and the Tensor Product Graph Method M. D. Gould and Y.-Z. Zhang
293
Free Field and Parafermionic Realizations of Twisted su(3yk ' Current Algebra X.-M. Ding, M. D. Gould and Y.-Z. Zhang
301
xxi
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1853-1865 © World Scientific Publishing Company
STAR-TRIANGLE EQUATIONS AND IDENTITIES IN HYPERGEOMETRIC SERIES
HELEN AU-YANG and JACQUES H.H. PERK Department of Physics, Oklahoma State University, 145 Physical Sciences, Stillwater, OK 74078-3072, USA
Received 1 January 2002 In this paper, we introduce the cyclic basic hypergeometric series p + i $ p with q —> u> where uiN = 1. This is a terminating series with N terms, whose summand has period N. We show how the Fourier transform of the weights of the integrable chiral Potts model are related to the 2 ^ 1 , which is summable. We show that 3^2 satisfies certain transformation formulae. We then show that the Saalschiitzian 4^3 series is summable at argument z = ui. This then gives the simplest proof of the star-triangle relation in the chiral Potts model. Finally, we let N —> 00, where the star-triangle equation becomes a two-sided identity for the hypergeometric series.
1.
Introduction
1.1.
Definitions
T h e g e n e r a l i z e d h y p e r g e o m e t r i c s e r i e s is d e f i n e d 1 ' 2 a s a
l)
p+lTp
---
J O-p+1 & ! , • • • ,bp
E
(ai)r
• • (aP+i)i
j (1)
(bi)i---(bP)ill
l=0
where (a)i = T(a + l)/T(a)
(2)
= a(a + 1) • • • (a +1 - 1),
w h i l e t h e b a s i c h y p e r g e o m e t r i c h y p e r g e o m e t r i c series is y
p+1
ai,---,ap+i
°°
p j3i,--,/3p
"
(<xi;q)r
•• (aP+i;
g)i
E(Pi\q)r--{Pr;q)i{q\q)i '
t
Z
(3)
l=0
in which , 1
, '
q)l
=
f (1 - x)(l
- xq) • • • (1 - xq1-1),
~ \ 1/[(1 - xq-^0.
- xq-*)
I > 0,
• • • ( ! - xq%
l < 0.
(4)
T h e h y p e r g e o m e t r i c series p + i F p c a n b e o b t a i n e d f r o m p + i $ p b y t a k i n g t h e l i m i t g -» 1
with
a = qa,
(a; q)i/(q; 1
q)i ->• (a)i.
(5)
1854 H. Au-Yang & J. H. H. Perk
1.2. Known
Identities
A hypergeometric series is summable if the series can be written in terms of ratios of products of Gamma functions, while for the summable basic series it is written in terms of the g-products defined in (4). The most well-known summation formula is due to Gauss 2F1
r(c)r(c-o-b)
a,b c
W
r ( c - a)T(c - b)'
which is a summation formula for 2F1 of unit argument. The other is Saalschutz's theorem 3^2
a, 6, —n c, d
(c-a)n(c-b)n \ ' , (c - a - b)n{c)n
for
c + d = a + b-n
+ l,
(7)
for a terminating Saalschiitzian 3F2 of unit argument. In general, a series is called Saalschutzian if it satisfies the Saalschiitz condition 1 + a\ H
h ap+i = b\ -\
\-bp.
(8)
Most of the summation formulae for the usual p + i F p hypergeometric series have basic series analogues. 1 ' 2 The summability condition on the argument of z — 1 for the hypergeometric series must be replaced by z = q for the basic series, while the Saalschiitz condition is seen from (5) to become qa1---ap+1=
Pi---/3p,
(9)
as a and b are the exponents of a and /3. As an example, Dougall's theorem summing a terminating 7F6 of unit argument generalizes to Jackson's theorem for terminating 8$7 of argument z = q. We shall now show that the basic hypergeometric series at root of unity are intimately related to the integrable chiral Potts model. Indeed, many of the results presented here are implicit in the earlier works. 3-13 Since the notations used in several of these works 6 - 1 1 are unconventional, making the connections obscure, we present here the results in more standard notation. 2. The Cyclic Basic Hypergeometric Series 2.1.
Definitions
Since most of the summation formulae are valid only for terminating series, it is straightforward to analytically continue q to a root of unity without any convergence problems. For q —> u> = e2m/N, we find (X;LJ)I+N
= (l-xN)(x;w)i, +
l
(u; u)i+N
= 0,
(10)
1
(11)
k
(12)
(x;u,U=UW V/[(-x) (vx- ;u;)l], (x; Lj)i+k = (x; ui)k {u x; u)i. 2
Star-Triangle Equations and Identities
1855
Prom (10), we see immediately that the basic series is ill-defined for q = ui unless a p + i = OJ~J for some J < N. We shall here restrict ourselves to the case with ap+i = CJ~N+1 = w, so that there are N terms in the series. Definition: A cyclic basic hypergeometric series is a terminating series of N terms P + I ^$=,P 1=0
(ai;uj)r--(ap;uj)i (/3i;w)/---(/3P;w)/
l
(13)
whose summand is periodic in N. Using (10), we find that the requirement for periodic summand is satisfied if JAAT ~N - : •F'T L '~ 1J r I-aJ
(14)
J=I
Unlike the ordinary basic hypergeometric series in (3), where the dependences on the parameters ai and Pi are elementary, the periodicity requirement makes the dependences on these parameters very complicated, with an extremely complex iV-sheeted Riemann sheet structure. Because of this periodicity, we may change the indices of the summation I -» —I in (13) and then let I —> I + k while using (11), to find P + I$^ ,P
uj,a±,-
Pi,---,PP
'
u>,cjf311,---,ui(3p1
P + I$^ «P
fc-'-Pp
(15)
z a i •••a > ua l i fc w, w a i , • • •, w ap (ai;Kj)f-(aP;^)t — p+i ^*p wk/3i,---,ujk(3p ;z ' J (ft ;w)fc---(/3p; «;)*,' l
p
k
(16)
Since a p +i = w, the series in (13) is called a Saalschutzian if w2a>ia2---ap
= ft/32 •••&,,
z = w.
(17)
Clearly, if the left-hand side of (15) is a Saalschutzian, so is the right-hand side, and vice versa. 2.2. Cyclic basic series
2*1
It has been found 3 ' 15 that the Boltzmann weights of the integrable chiral Potts model can be written in product form, i.e. W{n) = 7 r
(a;oj)n
(f3;uj)n
=
(JLYn (v/fc^-n
\jaJ
(w/a;w)-n
w " v
""
7" ' = ^1 - C a
(18)
The Boltzmann weight of an edge connecting spin a and spin b is chiral, namely, W(a — b) 7^ W(b — a), and arrows are introduced to indicate the direction from spin a to b, as shown in Fig. 1. Here we have introduced W*(a — b) = W(b — a) to indicate the operation of arrow-reversing. 3
1856
H. Au-Yang & J. H. H. Perk
Wm(a-b)
Wm(a-b) Fig. 1. Boltzmaim Weights
Since the weights are periodic with period N, their Fourier transforms may be written as JV-l
W{f\k)
unk W(n) =
= ^
2
$ i ui, a
k
(19)
0
n=0
2.2.1. Recursion Formula It has been found originally in Canberra 3 that
w(/>(0)
:
2
$ i
0
!7W
2
w.a
$ i
P
-l
57
(o;//3)fe(7;u;)fe (uja-f/P;ui)k
C9/7a;w)-* .(20) ak{u)/r,u))-k
The proof of this recursion relation has been given in our Taniguchi lectures. 4 This was later extended to a more general case by Kashaev et al. 7 2
$ i
ujn(i
;7W"
u>,a
i $ i
0
;7
{u/(3)k (jojkjn
(P;uj)n(i;Lj)k(uJa/P;uj)m-nL -(21) (a;uj)rn(uja'y/l3;uj)m-n+k
2.2.2. Baxter Formula Consider the determinant whose elements are the weights in (18), i.e. D=
det
W(l-k),
(22)
l
Baxter gave the following formula5 without proof: N-l
-1-3
/3) (l-w-i-j/3)(l-u}3a)_ [a — u>
(23)
where $Q
^
e Mr(N-l)(JV-2)/12JV_
4
(24)
Star-Triangle
Equations and Identities
1857
A detailed proof was subsequently given by us in Ref. 14. Since this is a cyclic determinant, we find N-l
N-l
j=0
W(f)(j)
j=l
(25)
W(/)(0)
Baxter's formula (23) and the recursion formula (20) may now be used to prove the following theorem. Theorem 1: Every cyclic basic hypergeometric series 2$i is summable, and is given by 2$1
i(iV-l)
U!
t ;T = u) N*$0(
-
p(u>a//3)p(i) p(a)p{<jjI(3)p{wa~f /(1)'
(26)
where t takes N different integer values for the N different Riemann sheets, and N-l
p(a)= l[{l-uja)^N.
(27)
Here summable mean that the series is expressible as products. It is worthwhile to emphasize that the basic hypergeometric function 2 $ i is an TV-valued function of a and /3 with a complicated Riemann surface. The function p(a) has N — 1 branch points at a = w-7 for j = 1, • • •, N — 1. Due to the appearance of the composite functions—particularly, ^(7) with 7 found from (18) to be an TV-valued function of a and /3—we can see that it is non-trivial to describe the Riemann surface. It is rather amazing even to us that Baxter and others (see Refs. 17,18 and citations quoted there) have somehow found a way out without the detailed knowledge of the Riemann surfaces. 2.3. Transformation
formula for
3
*2
We shall now derive a transformation formula for the cyclic basic series 3^2- Using the convolution theorem, we may write N-l 3$ ^2
u!,ai,a2
;7 (=0
(ft;?)i
.
i;q)i^J
N-l
£:
= iV"
1
^2$i
fc=0
' pi
;OJ
k
u 2$1
U),Ol2
W~7
0i
' ~
(28)
where .N
l - / 3 fN
1-/3// 1 vJV'
,N
Now we can use the recursion formula (20) to obtain 5
(29)
1858
H. Au-Yang & J. H. H. Perk
^ 2
w,ai,Q2
N~
\z
ft,& '
:u
i $ i
J $ I
w, z / u ,
w,a2
£
fo/otiu
oJQ-x
u)/u,u>a2z/(3iu'
(30)
Pi
Clearly, we may change /3 2 to Pi with u —> 7 1 , a n d o b t a i n a different transformation formula. If we let z = w in (30), t h e n it is easily seen t h a t t h e 3 $ 2 on the righth a n d side of the equation becomes a 2 $ i . As t h e t h r e e 2 $ i are s u m m a b l e , the cyclic hypergeometric series 3 $2 is also s u m m a b l e for a r g u m e n t z = u>. We conclude: T h e o r e m 2: Every cyclic basic hypergeometric series 3
2 has a transformation formula given by (30), and is s u m m a b l e for z — u>.
2 . 4 . The Saalschiitzian 2.4.1. Summation
4 * 3 and
the Star-Triangle
Relation
Formula
Consider a Saalschiitzian 4^3 for a r g u m e n t z = w, a n d use t h e convolution theorem t o express it as , $ c
U), W OJi, W 0 2 , u r c * 3
U}aP!,UJbp2,UcP3
:ui
N-l
N-1
oj,oj'-a3
£ 3 $
UJaP1,U»P2 ',UJ
fc=0
7
2^1
c
CJ
u p3
\-k
7 J '
(31)
where
7
N_{l-o$)
(1-^)
_ (i-/ff)(l-/ff) (1-ofXl-/^)
(32)
TW-
T h e first part of this equation a n d t h e Saalschutz condition oj2a\a2a3 m a y be used to solve 03 and also P3. T h i s gives a" =
1
T
JV
l-(aiQ27//3i/32)JV'
/3 3 = "°
=
CiT 0:10:20:3
Pip203,
(33)
frft
Now we use the transformation formula (30) for 3<&2 thrice, i.e. 1$2
0
6
w /?i,w /3 2
i7W
i3^2
u>,ujkj/u,ujb~apyu u, 7 J ,
AB3$2
w fc 7 2
wai a; 6 /3 2 '
w/7i,a;fe+1/7|' waai
(34)
ABC 3^2 where t h e first line is identical to (30) in which Oi = W0 2 //? 3 , Pi = P2 / « 3 ,
J o 2 =WOi//33, /3 2 = Pi/a3,
J 03 = LJ/P3 — P3 = u/al =
Wai/P2, Pi/a2,
(35)
Star-Triangle
Equations and Identities
1859
with superscript * denoting the arrow-reversing operation described in (18), and A = N~1
2 3>i
Lj,(Jjba2
2^1
wfc7
(36)
If we denote
if- 1-af'
7?
(37)
1-af'
{Ti*i)N
then by using (35) and (33), it is easy to verify that 7i=72/73>
(38)
72=7i/73,
which in turn can be used to show 7i72 = 7/7373 = 7 / V w a 3
leading to
(39)
73 = UJ/J = T ^ .
Using (29) in which z = 7, it is easy to verify that 1-{&V1/U)N
_C12\N
\-{Pi/ur
WJ
_ 7lN
1-{1/U)N
N
=(%Ul\
l7lj. . A T
= (
' i-(7/^ \irfx>
(40)
'
FVom (18) and (35), it follows that 7171 = w a i / f t . As a consequence, we may use the transformation formula (30) again to obtain the second equality in (34) with LJ,UJ
B = N~1 23>!
LJ,U)b-aP%/u
7/u
(Jj/U
7i
2^1
(41)
;7i
w,k+b-a^*0^7/u
Furthermore, from (37), we find that 1 - (1/7*)
/-*\JV (42) W l-(7i)' while from (35) and (18) we obtain QJIQ^ = Pijotx. Using the transformation formula (30) for the third time, we arrive at (34) with
N
1- m
C^N-1
2
$j
w/7:
;«i
2$1
fc+l/-*ia;
Q
(43)
2
Denoting i(ji) = 7t 7^
N~'
Wi(n) = Wi (-n) =
(44)
^JW
and using recursion formula (20) for 2^1 in these constants A, B and C, we find <$2
cj,cjaa1,u)°a2
k
W3(b-a)W*2(b-a) Wi(a)W2(6)
w-kbDa^\rMk^^ (7A/w«3;w) fc
, , , .b—ax,*
p:,* , ,1 — k
^-a/3*,/3*'
(45) 7
where D = [ABC}0 with a = 6 = c = fc = 0 i n (36), (41) and (43). Next, the recursion formula (20) is used to write 2
$i
oj,u) as w i-fe w c /3 3
=
Rwck(-y/33/uja3;uj)k DWz(c)a3k{T,u)k"
R D
Q
= 2
'
uj,a3
u>
. ft''
7"
(46)
1860
H. Au-Yang & J. H. H. Perk
Substituting these two equations into the convolution formula (31) and using a W{n + a) n(uJ a;uj)n = 7 W(a) (w«ft;w)„'
(47)
we find Wi{a)W2{b)Wi{c)i$3 N-lRWz(a
UaP1,UJbp2,0JCp3 iN-1N-1
~b)^2J2
ojk{c-b'l)W2(a
:u}\
=
''
- b - l)Wi{-l),
(48)
fc=0 1=0
where the summation over k can be carried out resulting in the delta function N6i,c-b- This then proves the following theorem: Theorem 3: Every cyclic Saalschiitzian basic hypergeometric series 4 $3 is summable for z = w. 2.4.2. Star-Triangle Relation Furthermore, (48) is also the star-triangle equation Y, Wx{a - d)W2(b - d)W3{c - d) = R W3(a - b)W2(a - c)Wi(b - c),
(49)
d=0
shown in Fig. 2. The weights W and W are defined in (44), in which the parameters cti, fa and on, fa are related by (35), while 7; and 7; are related by (38) and (39). These relations are very symmetric. C
Fig. 2.
Star-Triangle Relation, with 71 = 7273, 72 = 7 1 7 |
an
d 73 = 7 * 7 2 -
2.4.3. The Constant R The constant R was originally given in Ref. 3. A proof was published in Ref. 16 and their proof can also be used here. By defining the matrices G40M
= w^h ~ dl
G4»)cd = W3(c - d),
(Al)dtd. = Sd,d.W!(a - d),
Star-Triangle
(A)b,c = W!(b - c),
G42)c,c, = 5c,c,W2(a
- c),
Equations
and Identities
{Al)b,v = SbMW2{a
1861
- b), (50)
the star-triangle equation (49) may be expressed as (A2AtAl)b,c
= R{A%AiJ%)b,c
or
A2A\A\
= RA%MA%.
(51)
The determinants are also equal, i.e. JV-1
^=nt W (l)W (l) 0
WX{1)
2
det^2det^3
3
detAi
(52)
This gives the constant R in terms of determinants of matrices A defined in (50), which can be evaluated by Baxter's formula (23). Alternatively, R is seen from (46) to be a product of seven 2$i- It can also be evaluated using (26), which is much more tedious, and after many cancellations, this yields the same result. We have thus avoided the complexity in the Riemann surface by relegating it to the multiplicative constant R in the star-triangle equation. 2.4.4. Rapidity Lines To form commuting transfer matrices, it is necessary to assign rapidity lines to the weights. There are two possible weights shown in Fig. 1. There are two essentially different choices for the directions of the arrows. Original Choice By assigning the rapidity lines as we originally did in Ref. 3, also shown in Fig. 3, t h e n it is easily seen from Figs. 1 a n d 3 t h a t
r •«
Fig. 3.
pr v v 5
\W1(n)
= W1pr\P')-)
Original choice of t h e rapidities.
W2(n) = W2(n) =
Wqr(n), Wqr(n), 9
(W3(n) = W*pq(n), \ W3(n) = Wpq{n).
(53)
1862
H. Au-Yang
& J. H. H. Perk
This means « 1 = Ctpr,
f a2 = aqr,
( OJ3 =
u/J3pq,
Pi = Ppr,
} fa - 0qr,
< 03 =
u/apg,
71 = 7pr, « i = apr, A = /3pr,
[ 72 = %r, /• a 2 = a g r , •) #2 = Pqr,
[ 73 = 7pg, ( «3 = a p q , \ fa = Ppq,
71 = Ipr,
\12 = 7«r>
I, 73 = 7pg-
(54)
Consequently, we find from (35) OLpr — aqraPq, Ppr = PqrPpq/u),
( aqr = aprapq, \ (3qr = (3pr(3pq/w,
( apq = wctpr/ (3qT, \ /? p g =
(3pr/aqr.
(55)
From the relations for we see that we would like to have the products (Xqr&pq and (3qrfipq independent of q. For this to happen, we must have aqr and apq containing the same g-dependent factor, say xq, one in the denominator, the other in the numerator, such that the dependence on q cancels out upon multiplication. A similar reasoning holds for (3pq. In fact, we find the only choices are apq = Xq/xp,
$~Pq - u>yp/yq.
(56)
Using this in the second and third brackets of (55), we find apq = uxp/yq,
(3pq = vxq/yp.
(57)
It is easily verified that these choices satisfy the Saalschiitz condition in (17). The periodicity requirement on the argument at z — w u
= 717273 = J ! K1 - $JA1 - af)] ^
(58)
3=1
can be satisfied, if x?+V? = k(l + x?v?),
s=p,q,r.
(59)
Solving this equation for xs and substituting the solution into (37), we find Ipq = P-pVql^qVp Ipq = ^^p^pfJ-q/Vq,
/i s = (1 ~ fcxf )/&'.
This reproduces exactly the integrable solution found earlier.
(60)
3
Other Distinct Choice Only by flipping the directions of the middle rapidity line q, do we find a distinct arrangement of weights. This results in the equation w-i J ^ Wpr(a - d)Wrq(b - d)Wqp{c - d) = R Wqp{a - b)Wrq(c - a)Wpr{b - c). (61) d=0
Flipping other lines merely gives permutations of these rapidity lines in the two star-triangle equations, as can be seen from Fig. 1 and Fig. 4. To have the relation 10
Star-Triangle Equations and Identities 1863
q
P
\
c4
r -*
r-«
Fig. 4. Flipping the arrow of q. (35) to hold, we must have Oipr
=
UCtrq/ fjqp,
Ppr
==
PrqlQqpi
\P")
for which to be satisfied, we must choose a
pr
=
Ppr = 9pjr-
PJp9ri
\P")
2 3
The Saalschiitz condition in (17) yields ui p = 1. It is then easy to verify that it is not possible to find p, fs and gs satisfying condition (58). Since the relation (61) is more symmetric than (49) when comparing Fig. 3 with Fig. 4, the extra symmetry requirement on the weights makes a solution impossible in the present case. 3. The N -> oo Limits 3.1. Star-Triangle Identity
Equation as a Double-Sided
Hypergeometric
In the limit N —> oo, with a* = wai and /% = ubi, and allowing the spin n in (44) and thus the summation index d in (49) to run through all integers, we find 12,13 that if the Saalschiitzian condition (64)
ffli + 0-2 + a3 + 2 = b\ + b2 + b3. and condition resulting from (58) sin7rai sin7ro2 sin7ra3 = sin7r&i sin7r&2 sin7r&3,
(65)
are satisfied, the star-triangle equation (49) becomes
£
( a l ) n n + n ( a 2 ) m 2 + n Mms+n
(h)mi+n
n=—oc \ i-'mi+n
(h)m2+n v-"/r
f ai = 1 + a 2 \bi = b2-a3
(b3)m3+n
RD0
( 5 l ) m i - m 2 (°-?)m2-m3
(h
( a2 = 1 + a\ - 63, \ b2 - h - as, 11
(62
i^r,
(bs) m.i—m.3
f a 3 == 1 + !ai - 62, \ 63 = bi - o 2 .
L
, (66)
(67)
1864 H. Au-Yang & J. H. H. Perk 3.1.1. Double-Sided Hypergeometric
Identity
The equation (66) may be rewritten as the double-sided summation identity 13 V* TT r ( a » + n) _ g ( a i > a 2 , a s \ b \ , b 2 , h )
. ,
where TT5
G(ai,a 2 ,a 3 |6i,62,6 3 ) = — sin7ra2 sin7ra3 rii=i sin7r(6i —ai) 3 3
= J J T{aj)Y{l - aj) Y[T{bi - ai)r(l - h + 01), j=i
(69)
i=\
provided the two conditions (64) and (65) are satisfied. If we let aj —> a% + m and bj —> aj + m, then these two conditions are still satisfied. This shows that the above two-sided identity holds for infinitely many different values of a, and bj and is rather unusual. 3.2. Its Dual (Fourier
Transform)
If, instead of demanding that the spin values n in (44) and d in (49) remain integers, we let d,n,N-^ 00, while keeping the ratios y = 2im/N and x = 2ird/N finite, we find that the summation over N values in (49) becomes an integral over the interval [0,27r]. More specifically, we find the weight (18) to become 13 TI// u i / s i n 7 r 6 \ ( ^ ~ L^J) I . ! ,a-b W(a,b,x) = [sinix . \sm7ra/ 1 ^ 1
,_„> (70)
in this limit. For a* and 6j satisfying the two conditions (64) and (65), we let Wi(x) = W(ai, bi, x),
Wi(x) = W(ai, bu x),
and the star-triangle relation (49) becomes
(71)
13
1 /"27r —- / dw Wi(x — w) W?{y — w) W^{z — w) 27T J0
= R00W3{x-y)W1{y-z)W2{x-z).
(72)
Since the weights are chiral, namely, W(—x) ^ W(x), it is not possible to have both the weights and their Fourier transforms real. Thus the Fourier transform of (72) is an identity similar to (66), but not identical, and vice versa. 3.3. Open
Problems
Finally, the weights in Sections 3.1 and 3.2 define integrable models, which are limiting cases of the original chiral Potts model, and are chiral extensions of the models in the works of Fateev and Zamolodchikov. 19 ' 20 Since there are sets of N functional relations for the chiral Potts models, we expect there may then be infinitely many such functional relations for these models, and perhaps some more physical quantities in these oo-state models can be evaluated. 12
Star-Triangle Equations and Identities
1865
Acknowledgments We t h a n k Professors Yuri Stroganov and George Andrews for helpful discussions. T h e hospitality and h a r d work by the local organizing committee a t the Nankai I n s t i t u t e is greatly appreciated. This work has been supported in p a r t by NSF G r a n t s No. PHY-95-07769, P H Y 97-22159 and PHY 97-24788 and P H Y 01-00041.
References 1. W. N. Bailey, in Generalized Hypergeometric Series, (Stechert-Hafner, New York and London, 1964), p. 65-70. 2. L. J. Slater, in Generalized Hypergeometric Functions, (Cambridge University Press, Cambridge, 1966). 3. R. J. Baxter, J. H. H. Perk, and H. Au-Yang, Phys. Lett. A 128, 138-142 (1988). 4. H. Au-Yang and J. H. H. Perk, in Integrable systems in quantum field theory and statistical mechanics, Advanced Studies in Pure Mathematics, Vol. 19, M. Jimbo, T. Miwa and A. Tsuchiya, eds. (Kinokuniya-Academic, Tokyo, 1989), pp. 57-94. 5. R. J. Baxter, J. Stat. Phys. 52, 639-667 (1988). 6. V. V. Bazhanov and R. J. Baxter, J. Stat. Phys. 69, 453-485 (1992), 71, 839-864 (1993). 7. R. M. Kashaev, V. V. Mangazeev, and Yu. G. Stroganov, Intern. J. Mod. Phys. A 8, 587-601 (1993), A 8, 1399-1409 (1993). 8. V. V. Mangazeev, S. M. Sergeev, and Yu. G. Stroganov, Intern. J. Mod. Phys. A 9, 5517-5530 (1994); Mod. Phys. Lett. A 10, 279-287 (1995). 9. S. M. Sergeev, V. V. Mangazeev, and Yu. G. Stroganov, J. Stat. Phys. 82, 31-49 (1996). 10. Z. N. Hu, Intern. J. Mod. Phys. A 9, 5201-5213 (1994); Phys. Lett. A 197, 387-392 (1995). 11. Z. N. Hu and B. Y. Hou, J. Stat. Phys. 79, 759-764 (1995). 12. H. Au-Yang and J. H. H. Perk, Int. J. Mod. Phys. (Proc. Suppl.) 3A, 430-434 (1993). 13. H. Au-Yang and J. H. H. Perk, Physica A 268, 175-206 (1999). 14. H. Au-Yang and J. H. H. Perk, Int. J. Mod. Phys. B 11, 11-26 (1997). 15. H. Au-Yang, B. M. McCoy, J. H. H. Perk, and S. Tang, in Algebraic Analysis, Vol. 1, M. Kashiwara and T. Kawai, eds. (Academic, San Diego, 1988), pp. 29-40. 16. V. B. Matveev and A. O. Smirnov, Lett. Math. Phys. 19, 179-185 (1989). 17. R. J. Baxter, V. V. Bazhanov, and J. H. H. Perk, Intern. J. Mod. Phys. B 4, 803-870 (1990). 18. H. Au-Yang, B.-Q. Jin, and J. H. H. Perk, J. Stat. Phys. 102, 471-499 (2001). 19. V. A. Fateev and A. B. Zamolodchikov, Phys. Lett. A 92, 37-39 (1982). 20. A. B. Zamolodchikov, Phys. Lett. B 97, 63-66 (1980).
13
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1867-1873 © World Scientific Publishing Company
H A P P E R ' S CURIOUS DEGENERACIES A N D Y A N G I A N
CHENG-MING BAI Liuhui Center for Applied Mathematics and Theoretical Physics Division of Nankai Institute Mathematics, Nankai University, Tianjin 300071, P.R. China
of
MO-LIN GE Liuhui Center for Applied Mathematics and Theoretical Physics Division of Nankai Institute Mathematics, Nankai University, Tianjin 300071, P.R. China
of
KANG XUE Liuhui Center for Applied Mathematics and Theoretical Physics Division of Nankai Institute of Mathematics, Nankai University, Tianjin 300071, P.R. China Dept. of Physics, Northeast Teaching University, Chang Chun, Jilin, 130024, P.R. China
Received 27 November 2001
We find raising and lowering operators distinguishing the degenerate states for the Hamiltonian H = x(K + \)SZ + K • S at x = ± 1 for spin 1 that was given by Happer et a l . 1 ' 2 to interpret the curious degeneracies of the Zeeman effect for condensed vapor of 8 7 R b . The operators obey Yangian commutation relations. We show that the curious degeneracies seem to verify the Yangian algebraic structure for quantum tensor space and are consistent with the representation theory of Y(sl(2)).
1. Indecomposible Quantum Tensor Space In Quantum Mechanics, a state is described in terms of wave function, i.e. \ip > is a vector in Hilbert space. If two particles described by \tp12 > are entangled, there should be "overlapping effect" between V\ and V2, i.e., besides V\ and V2 we should deal with V\ V2, the quantum tensor space. The simplest example is Breit-Rabi's Hamiltonian: HBR = K-s +xks3,
(1.1)
where s and K stand for the spins of electron and atomic nucleus, respectively. K 2 = K(K + 1). On account of the conservation of K 2 and m = K% + S3 two independent states are introduced: a1>=\K,m-->\-,->,
a2>=\K,m+->\-,-->. 15
(1.2)
1868
C.-M. Bai, M.-L. Ge & K. Xue
For a fixed m with the basis 3> = I + \l(xk
# B R =~\ where u\ = I
), we have +mV3 + x/^-mVi],
I and 03 = I
diagonalized through a rotation: Ui^H^U^r1
(1.3)
J are Pauli matrices. Eq. (1.3) can be
1
= fl&W). &m\
(1-4)
where ^mj_V(sin^)|ai>+(cos^)|a2>j' E=---uma3,
{1
*>
(1.6)
and cos<^m = ^
^ ,
w^ = (1 + x2)k2 + 2xmk.
(1.7)
Noting that the rotation angle ipm is m-dependent and m here cannot be replaced by the operator Kz + S3. This is because of the nonlinearity in m, i.e., the rotation should depend on the history. Observing Eq. (1.6) and Eq. (1.7) there is not degeneracy for the energy E, because the vanishing wm means a complex magnetic field. However, there appears degeneracies for spin-1 in the experiment. 2 Why the Zeeman effect vanishes at the particular value of applied field? This is the main subject concerned in this paper. 2. Introduction of the Curious Degeneracies The curious degeneracies observed in the experiment for condensed vapor of 8 7 Rb and 8 5 Rb x at 220° under pressure and applied magnetic field B ~ 1500 Gauss are converted into "anti-level-crossing" for the triplet {S = l ) . 1 , 2 To describe the Hamiltonian of a triplet dimer neglecting the quadrapole interaction, Happer et al. introduced 1 ' 2 H = K-S
+ x(K + ±)Sz,
(2.1)
and pointed out that when x = 1 there appear the curious degeneracies for S = 1, where K and S are angular momentum and spin, respectively, K 2 = K(K +1) and S 2 = S(S + 1) with 5 = 1. In Ref. 1, the eigenvectors corresponding to E = — | had been given and an elegant discussion was made. However, there remain the following essential questions: • Why the curious degeneracies occur only for S = 1? 16
Happer's
Curious Degeneracies and Yangian 1869
• How to distinguish the degenerate states? We would like to present the answer in this paper. For x = ± 1 , the eigenequation HVm
= EmVm
(2.2)
has three types of solutions whose eigenstates are denoted by ax, OLD and OLB with the corresponding energies ET > ED > EB, respectively. For the .D-set, Ha Dm = — \&Dm, there appear the curious degeneracies called Happer degeneracies that has been supported by the experiment. 2 The results of Happer can be summarized in the Table 1 (G = K + S, G3 = m). G= K+l
G= K
G= K-l
K+l K K - 1
-> -> ->
m
->
-K
+l ~K - K - l
D-set
T - set
<*D,m=K OD,ra=K-l
<XT,m=K+l °
<*B,m=K-\
OiTm
OLBm
<*T,m=-K+l CtT,m=-K
«fl,m=-Jf+l <*B,m=-K
ftDm
- > • Q-D,m=-K+\ -¥ ->• <XD,m=-K-\
B - set
Table 1 We emphasize that the states with m = K + 1 and m — -K for x = 1 (m = —K—l and m = K for x = — 1) in the D-set are excluded. For simplicity we discuss the case for x = 1 henceforth. The eigenstates of H are linear combinations of the states of G = K + 1, K and K — l. Since the shortage of states with m = K + 1 and m = —K it is not surprise to appear the unusual thing to distinguish the m-dependent states, for example in Eq. (1.6). 3. Yangian as the Raising and Lowering Operator for the Degenerate States Let us first recall how to establish the Lie algebraic structure in Quantum Mechanics. For the given (2K +1) states denoted by \K, K3 = K >, \K, K3 = K - 1 >,• • •, and \K,K3 = — K > , the raising (or lowering) operator K+ (or K_) can be introduced such that for any m = K3, K±\K,m>~\K,m±l>,
(3.1)
and K ± \K, +K > = 0. Through checking the commutation relations for K± and K3, we say that the Lie algebraic structure is found if the commutation relations are closed. It is emphasized that there is not m-dependence in the operators K± in Eq. 17
1870
C.-M. Bai, M.-L. Ge & K. Xue
(3.1), because the eigenvalues of K3 are uniform. However, suppose the eigenvalues are not uniform, the raising and lowering operators should depend on m, i.e., it should indicate on which state the operators act. Actually, such "starting state" dependence occurs more often in nonlinear models.3 After calculations, we have found the raising operator for the D-set in the table 1 (at x = ±1): J+ = (m + K + l)G++j+(a,b),
(3.2)
where j+(a, b) = aS+ + bK+ + i ( 5 3 X + - S+K3),
(3.3)
a = -j,b-a=±(K
(3.4)
and + l),G+=K+
+ S+.
Noting that (b — a) is independent of m. Whereas J-=-(m
+ K)G- + L (c, d),
(3.5)
where j'_(c,d) = cS- + dK^ - i ( 5 3 l f - - S-K3),
(3.6)
c=j + ±,d-c=-j,G.=K-+S-.
(3.7)
and
It can be checked that for x = 1, J+\aD,m=K > = 0 and J+\aD,m=-K-i > = 0. Obviously the J± shown in Eq. (3.2) and Eq. (3.5) are special form of the Yangian operator: J-AG+j,
(3.8)
j = /iK + 7 S - ^ S x K ,
(3.9)
where
and A,/i, 7 are arbitrary constants. A set formed by both J and j satisfy defined by Drinfeld,4 and is related to the Yang-Baxter equations. 5,6
Y(sl(2))
4. Yangian Algebra The commutation relations for J and the total angular momentum I = G = S + K form the so-called Yangian algebra associated with s/(2). The parameters /i and 7 play the important role in the representation theory of Yangian given by Chari and Pressley. 7 Many chain models possess the Yangian symmetry, for example, for 1-d Hubbard model and Haldane-Shastry model. 8 The set {I, J } = Y(s/(2)) obeys the commutation relations of Y(sl(2)) (A± = A\ ± 1/^X^2): [h, I±] = ± / ± , [/+, / - ] = 2/3, 18
(«l(2));
(4.1)
Happer's Curious Degeneracies and Yangian
[h, J±] = [J3, /±] = ±J±,
[/+, J-} = [J+, I-] = 2J3,
1871
(4.2)
(i.e. [Ii, Jj] = -\/—Teijfe Jfc) and nonlinear relation [J 3 , [J+, J-}] = \h(I+J-
- J+I-)
(4.3)
that forms an infinitely dimensional algebra. All the other relations given in Ref. 4 can be obtained from Eq. (4.1)-Eq. (4.3) together with the Jacobian identities. 9 ' 10 The essential difference between the representations of Yangian algebras and those of Lie algebras is the appearance of the free parameters fj, and 7 whose originally physical meaning is one-dimensional momentum. Their special choice specifies a particular model. Applying the Yangian representation theory to Hydrogen atom, it yields the correct spectrum (~ n~2) that is the simplest example of the application of Yangian in Quantum Mechanics. 10 Now the Happer's degeneracies can be viewed as another example. Furthermore, we would like to make the following remarks: (a) The elements of J+ given by Eq. (3.2) < aDm'\J+\o.Dm > ~ < aDm>\K+\aDm
> ^ 0,
because < aDm'\S\aDm >=< (*Dm'\S x K.\aum > = 0, as pointed out in Ref. 1 (see Eq. (2.23) in Ref. 1). This indicates that the role played by J+ in the ".D-direction" is like that played by K+. Why do we need a Yangian? The terms of 5+ and (K x S ) + should be added to guarantee < oiTm'\J+\<XDm >=< aBm'\J+\aDm >= 0, namely, if only acting K+ on a n m it yields non-vanishing transitions to arm' and a.Bm' that no longer preserves the D-set. The part other than K+ in the Yangian J + given by Eq. (3.2) exactly cancel the nonvanishing contribution received from "T-" and "B-direction". (b) Observing the process determining parameters a and b in Eq. (3.3), the reason for the existence of solution of a and b is clear. For 5 = 1, the eigenvector of H is formed by three base. Apart of an over-all normalization factor there are two independent coefficients. In requiring J+a.r>m ~ a c r a + i , we have to compare the coefficients of the independent base in J+ar>m and aom+i to determine the unknown parameters a and b. For spin 5 = 1 , there are just two equations for a and b. However, for spin 5 > 1, in general, one is unable to find solution for a and b to fit more than two equations. Therefore, the Yangian description of the curious degeneracies admits only 5 = 1 for arbitrary K. This is consistent with experiment. 1 ' 2 (c) In fact, the parameters appearing in J+ and J_ exactly coincide with the conditions of the existence of the subrepresentations of the Yangian. 7 Following the theorem in Ref. 7, for a — b = — ^ — 5 the subspace spanned by vectors with G = K + 1 is the unique irreducible subrepresentation of Y(sl(2)), that is, the states with G = K + 1 are stable under the action of J. Note that the existence and uniqueness of subrepresentation is only related to the difference of a and b. Moreover, for the given a and b in Eq. (3.4), the action of J+ on the states with 19
1872
C.-M. Bai, M.-L. Ge & K. Xue
G = K + 1 is given by J+aa=K+i,m = (m + K + l)G+aG=K+i,m and at the same time, J+ will make the states with G — K and G = K - 1 transit to G = K + 1, but not vice versa, called "directional transition", 9 i.e. the transition given rise by Yangian goes in one way. Thus, for the given a and b in Eq. (3.4), the set of states with G = K + 1 and D-set are stable under the action of J+ simultaneously. For c — d = ^, G = K — 1 is the unique irreducible subrepresentation and for c and d given by Eq. (3.7), acting J_ on the states with G = K — 1, we have J-CtG=K-\,m = —(m + K)G~aG=K-i,m- Therefore the representation theory of Y(sl(2)) tells that the relationship between a — b and c - d given by Eq. (3.4) and Eq. (3.7), respectively, should be held to preserve the states with G = K + 1 (or G = K — 1) that possesses Lie algebraic behavior. (d) We have seen that the J_ is not the conjugate of J+. Such a phenomenon is reasonable because ar>m is neither the Lie-algebraic state nor symmetry of H. In fact, if a is not an eigenstate of I 2 (I belongs to a Lie algebra) and I+a ~ a i , we cannot have i _ a i ~ a. Now there is the similarity for Yangian. Moreover, the -D-set is not a subrepresentation of Y(sl(2)), i.e., D-set cannot be stable under all the actions of J, but stable under J+ and J_ with the different parameters which just satisfy the condition for subrepresentation of Yangian. (e) The third component of J takes the form J3 = aSz + bKz + S+K- — S-K+. For any parameters, the action of J3 will not keep the D-set. But, with the suitable a — b = 1, the operator J 3 + 2(2K + 1)S% will keep the .D-set. (f) We emphasized that the m appearing in Eq. (3.2) and Eq. (3.5) cannot be replaced by the operator G3. It appears as a parameter in Yangian. The m-dependents only indicates that the raising or lowering operation depends on "history" in difference from the Lie algebraic structure. In conclusion we have read of a new type of algebra structure (Yangian) from the Happer's degeneracies and such an algebra had been ready by Drinfeld.4 All the analysis coincides with the representation theory of Y(sl(2))7 for the special choice of o, b in J+ and c, d in J_. It also leads to the fact that only S = 1 is allowed to yield the curious degeneracies. If the Zeeman effect tells Lie algebra, then the curious degeneracies possibly tell the existence of Yangian. Acknowledgement s We would like to thank Freeman Dyson who guided the authors to deal with this problem and W. Happer for encouragement and enlightening communications. This work is in part supported by NSF of China. References 1. W. Happer, Degeneracies of the Hamiltonian x(K+l/2)Sz+K-S, preprint, Princeton University, November, 2000. 2. C.J. Erickson, D. Levron, W. Happer, S. Kadlecek, B. Chann, L.W. Anderson, T.G. Walker, Phys. Rev. Lett. 85, 4237 (2000). 20
Happer's Curious Degeneracies and Yangian
1873
3. J. L. Chen and M.L. Ge, Phys. Rev. E60, 1486 (1999). 4. V. Drinfeld, Sov. Math. Dokl. 32, 254 (1985); 36, 212 (1988); Quantum groups, in Proc. ICM, Berkeley, 269 (1986). 5. L.D. Faddeev, Les Houches, Session 39, 1982. 6. E.K. Sklyanin, Quantum Inverse Scattering Methods, Selected Topics, in M.L. Ge (ed.) Quantum Groups and Quantum Integrable Systems, World Scientific, Singapore, 6388 (1991). 7. V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, 1994; Yangian and R-matrix, L'Enseignement mathematique 36, 267 (1990). 8. D.B. Uglov and V. Korepin, Phys. Lett. A190, 238 (1994); F.D.M. Haldane, Phys. Rev. Lett. 60, 635(1988); S. Shastry, Phys. Rev. Lett. 60, 639 (1988); Haldane F.D.M., Physics of the ideal semion gas: Spinons and quantum symmetries of the integrable Haldane-Shastry spin chain, Proceedings of the 16th Taniguchi Symposium on Condensed Matter Physics, Kashikojima, Japan, ed. by O. OKiji and N. Kawakami, Springer, Berlin, 1994. 9. M.L. Ge, K. Xue,and Y.M. Cho, Phys. Lett. A249, 258 (1998); C M . Bai, M.L. Ge and K. Xue, Directional Transitions in spin systems and representations of Y(sZ(2)), Nankai preprint, APCTP-98-026. 10. C M . Bai, M.L. Ge and K. Xue, J. Stat. Phys. 102, 545 (2001).
21
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1875-1882 © World Scientific Publishing Company
F R A C T I O N A L STATISTICS IN SOME EXACTLY SOLVABLE MODELS W I T H P T INVARIANT INTERACTIONS
B. BASU-MALLICK Saha Institute of Nuclear Physics, Theory Group 1/AF Bidhan Nagar, Calcutta 700 064, India E-mail; [email protected]
Received 19 November 2001 Here we review a method for constructing exact eigenvalues and eigenfunctions of a many-particle quantum system, which is obtained by adding some nonhermitian but P T invariant (i.e., combined parity and time reversal invariant) interaction to the Calogero model. It is shown that such extended Calogero model leads to a real spectrum obeying generalised exclusion statistics. It is also found that the corresponding exchange statistics parameter differs from t h e exclusion statistics parameter and exhibits a 'reflection symmetry' provided the strength of the P T invariant interaction exceeds a critical value.
1. Introduction It is well known that integrable dynamical models and spin chains with long range interactions exhibit fractional statistics or generalised exclusion statistics (GES), 1 which is believed to play an important role in many strongly correlated systems of condensed matter physics. The A^v-i Calogero model (related to A^-i Lie algebra) is the simplest example of such dynamical model, containing N particles on a line and with Hamiltonian given by Refs. 2,3 1 v^ °
w
v^ 2
9 v^
1
+
tf = - 2 E ^ TE*KE(^r
(1)
where g is the coupling constant associated with long-range interaction. One can exactly solve this Calogero model and find out the complete set of energy eigenvalues as Enun2,-,nN
N = -—[l
N
+ (N-l)v]+u^2nj.
(2) j=l
Here rijS are non-negative integer valued quantum numbers with rij < nj+i and v is a real positive parameter which is related to g as ,2
9 = v*-v. 23
(3)
1876
B.
Basu-Mallick
It may be noted that, apart from a constant shift for all energy levels, the spectrum (2) coincides with that of N number of free bosonic oscillators. Furthermore, one can easily remove the above mentioned constant shift for all energy levels and express (2) exactly in the form of energy eigenvalues for free oscillators: Eni where fij = rij + v(j — 1) are quasi-excitation numbers. However it is evident that these fijS are no longer integers and they satisfy a modified selection rule given by fij+i — fij > v, which restricts the difference between the quasiexcitation numbers to be at least v apart. As a consequence, the Calogero model (1) provides a microscopic realisation for fractional statistics with v representing the corresponding GES parameter. 4 ' 7 Recently, theoretical investigations on different nonhermitian Hamiltonians have received a major boost because many such systems, whenever they are invariant under combined parity and time reversal (PT) symmetry, lead to real energy eigenvalues. 8-11 This seems to suggest that the condition of hermiticity on a Hamiltonian can be replaced by the weaker condition of PT symmetry to ensure that the corresponding eigenvalues would be real ones. However, till now this is merely a conjecture supported by several examples. Moreover, in almost all of these examples, the Hamiltonians of only one particle in one space dimension have been considered. Therefore, it should be interesting to test this conjecture for the cases of nonhermitian iV-particle Hamiltonians in one dimension which remain invariant under the PT transformation: 12 i —• —i,
Xj —• —Xj,
Pj - 4 pj ,
(4)
where j £ [1,2, • • •, N], and Xj (pj = —igfr) denotes the coordinate (momentum) operator of the j-th particle. In particular, one may construct an extension of Calogero model by adding to it some nonhermitian but P T invariant interaction, and enquire whether such extended model would lead to real spectrum. The purpose of the present article is to review the progress 7 ' 1 2 on the above mentioned problem for some special cases, where the P T invariant extension of Calogero model can be solved exactly. In Sec.2 of this article we consider such a PT invariant extension of AN-I Calogero model and show that, within a certain range of the related parameters, this extended Calogero model yields real eigenvalues. Next, in Sec.3, we explore the connection of these real eigenvalues with fractional statistics. Section 4 is the concluding section. 2. Exact solution of an extended Calogero model Let us consider a Hamiltonian of the form7
U = H + SY——A &k
Xj
Xk
dXj
(5)
where H is given by eqn.(l) and 5 is a real parameter. It may be observed that though the Hamiltonian (5) violates hermiticity property due to the presence of 24
Fractional Statistics in Some Exactly Solvable Models 1877
momentum dependent term like S £V; ,fe x \ x -^-, it remains invariant under the combined P T transformation (4). Next we recall that, .AJV-I and B^ Calogero models as well as their distinguishable variants have been solved recently by mapping them to a system of free oscillators.13"16 With the aim of solving the PT invariant extension (5) of A/v_i Calogero model through a similar produce, we assume that (justification for this assumption will be given later) the ground state wave function for this extended model is given by
j
where v is a real positive number which is related to the coupling constants g and 6 as g = v2-v(l
+ 26).
(7)
Now if we use the expression (6) for a similarity transformation to the Hamiltonian (5), it reduces to an 'effective Hamiltonian' of the form W = ^-Hi)gr
= S-+ujS3
+ Egr,
(8)
where the Lassalle operator (S~) and Euler operator (S 3 ) are given by
J= l
J
Jjik
J
J
3= 1
J
and Egr = ^ \ \ + {N-l){v-6)].
(10)
It is easy to see that the Lassalle operator and Euler operator, as defined in eqn.(9), satisfy the simple commutation relation: [S3, S~] = —2S~. Using therefore the well known Baker-Hausdorff transformation we can remove the S~ part of the effective Hamiltonian H' and through some additional similarity transformations reduce it finally to the free oscillator model7
j=l
where 5 = e^s
e^e*£"=i
j
j=l
^ and V 2 = Y%=i 8*
dx'i
Due to similarity transformations in (8) and (11), one may naively think that the eigenfunctions of the extended Calogero model (5) can be obtained from those of free oscillators as: , 0 ni! „ 2i ... ) „ Af = ipgrS < Ylj=i e~^XjHnj{xj) >, where rijS are arbitrary non-negative integers and Hnj(xj) denotes the Hermite polynomial of order rtj. However it is easy to check that, similar to the case of A^-i Calogero model, 13 the action of «S on free oscillator eigenfunctions leads to a singularity unless they are symmetrised with respect to all coordinates. Therefore, nonsingular eigenfunctions 25
1878
B. Basu-Mallick
of the extended Calogero model (5) can be obtained from the eigenfunctions of free oscillators as V'm.ns.-.nAT = i>gr SA+ I J \ ^~^Hnj(Xj)
>,
(12)
where A + completely symmetrises all coordinates and thus projects the distinguishable many-particle wave functions t o the bosonic part of the Hilbert space. Evidently, the eigenfunctions (12) will be mutually independent if the excitation numbers TijS obey the bosonic selection rule: rij+i > rij. Thus, in spite of the fact that the interacting Hamiltonian (5) is convertible to the free oscillator model, the need for symmetrization shows that the many-particle correlation is in fact inherent in this model. The eigenvalues of the Hamiltonian (5) corresponding to the states (12) will naturally be given by Ref. 7 N n
Enun2,-,nN =Egr+UjJ2 J
N
N 1
= ^fl
+ (N-
l)(v-5)}+LjJ2nJ-
j=l
(13)
j=l
Since d and v are real parameters, the energy eigenvalues (13) are also real ones. Thus we interesting find that the nonhermitian P T invariant Hamiltonian (5) yields a real spectrum. Furthermore, it is evident that for all rij = 0, the energy Eni,n2,—,nN attains its minimum value Egr. At the same time, as can be easily worked out from eqn.(12), the corresponding eigenfunction reduces t o if)gr (6). This proves that ^)gr is indeed the ground state wave function for Hamiltonian (5) with eigenvalue Egr. It may be observed that the eigenfunctions (12) pick up a phase factor (—1)" under the exchange of any two particles. Therefore, v represents the exchange statistics parameter for the extended Calogero model (5). By solving the quadratic eqn.(7), one can explicitly write down v as a function of g and 6 as v
= {5+l-)±yjg + {8+\f
(14)
For the purpose of obtaining real eigenvalues (13) as well as nonsingular eigenfunctions (12) at the limit Xi —• Xj, we have assumed at the beginning of this section that v is a real positive parameter. This assumption leads to a restriction on the allowed values of the coupling constants g and 6 in the following way. First of all, for the case g < — (6 + \)2, eqn.(14) yields two imaginary solutions. Secondly, for the case 6 < - \ , 0 > g > -(5 + \)2 eqn.(14) yields two real but negative solutions. Inequalities corresponding to these two cases represent two forbidden regions of ((5, g) plane which are excluded from our analysis. For the case g > 0 with arbitrary value of S, one gets a real positive and a real negative solution from eqn.(14). The real positive solution evidently leads to physically acceptable set of eigenfunctions and corresponding eigenvalues within this allowed region of (8, g) plane. Finally we consider the parameter range 6 > 26
Fractional Statistics in Some Exactly Solvable Models 1879
— \, 0 > g > — (6 + \)2, for which eqn.(14) yields two real positive solutions. It is easy to see that these two real positive solutions are related to each other through a 'reflection symmetry' given by v —> 1 + 26 - v. Consequently, for each point on the (S,g) plane within this allowed parameter range, one obtains two different values of the exchange statistics parameter leading to two distinct sets of physically acceptable eigenfunctions and eigenvalues. Thus we curiously find that a kind of 'phase transition' occurs at the line 6 = — | on the (S,g) plane. For the case 6 > — | , exchange statistics parameter shows the reflection symmetry when g is chosen within an interval — ( | + 6)2 < g < 0. On the other hand for the case 6 < — ^, such reflection symmetry is lost for any possible value of g. We have seen in this section that, similar to the case of Aw-i Calogero model, the extended model (5) can also be solved by mapping it to a system of free harmonic oscillators. So it is natural to enquire whether this extended model is directly related to the AM-i Calogero model through some similarity transformation. Investigating along this line, 12 we find that
r - ^ ^ ' ^ ^ +^ E ^ + s ' E ^ j i .
(is)
where T = Y\j 0, g > —6(1 + 6), there exists a positive solution of eqn.(7) satisfying the condition v — 6 < 0. Therefore, we can not get any lower bound for the corresponding energy eigenvalues (13) at N -> oo limit. On the other hand, the energy eigenvalues (2) of >1AT-I Calogero model are certainly bounded from below for all possible choice of N and g. So there exists a parameter range within which the spectrum of extended Calogero model differs qualitatively from the spectrum of the original Calogero model. To explain this rather unexpected result, we first observe that the renormalised coupling constant g' would be a positive quantity within the above mentioned parameter range. Consequently, the corresponding exclusion statistics parameter v', which is obtained by solving eqn.(3), has one positive and one negative solution. One usually throws away this negative solution of v'', since the corresponding eigenfunctions become singular at the limit Xj —> Xk- However, by using the relation (15), such singular eigenfunctions (denoted by ip'(x\,X2, • • • ,£JV)) may now be used to construct the eigenfunctions of extended Calogero model (denoted by i[}(xi,X2, • • • XN)) as ip(x1,x2,---,xN)
=
J\(XJ -xk)Sip'(x1,x2,---,xN).
(16)
3
It can be easily checked that, due to the existence of the factor Ylj xk • Thus we curiously find that singular eigenfunctions of H' can be used to generate nonsingular 27
1880
B. Basu-Mallick
eigenfunctions of H through the relation (16). This shows that the similarity transformation (15) is a subtle one and, within a certain parameter range, eigenvalues of extended Calogero model will match with those of Calogero model (having renormalised coupling constant) only if the corresponding unphysical eigenfunctions are taken into account. 3. Connection with fractional statistics We have mentioned in Seel that GES can be realised microscopically in AN-I Calogero model with hermitian Hamiltonian. The GES parameter for this Calogero model is a measure of 'level repulsion' of the quantum numbers generalising the Pauli exclusion principle. Now for exploring GES in the case of PT invariant model (5), we observe that eqn.(13) can be rewritten 7 exactly in the form of energy spectrum for N free oscillators as Eni,n3-nN
N iv = - 5 - -t-W^flj-,
(17)
where ni=ni + (v-8)(j-l)
(18)
are quasi-excitation numbers. However, from eqn.(18) it is evident that such quasiexcitation numbers are no longer integers and satisfy a modified selection rule: fij+i — fij > v — 5. Since the minimum difference between two consecutive fijS is given by v = u-S,
(19)
the spectrum of extended A^-i Calogero model (5) satisfies GES with parameter v.7 Several comments about this GES parameter are in order. It may be to noted that for 6 ^ 0, the GES parameter v is different from the power index v, which is responsible for the symmetry of the wave function. Therefore we may interestingly conclude that unlike Calogero model, the exclusion statistics for model (5) differs from its exchange statistics. Furthermore it is already noticed that, on a region of (6,g) plane satisfying the inequalities 6 > 0, g > -5(1 + 8), there exists a positive solution of eqn.(7) which yields a negative value of v. For this case, however, one does not get well defined thermodynamic relations at N —> 00 limit and, therefore, can not interpret v as the GES parameter. By using eqn.(7) and (19), we find the relation i>2-v = g + 8(8 + l),
(20)
which clearly describes a parabolic curve in the coupling constant plane (6, g) for any fixed value of v. As a consequence of this, the competing effect of the independent coupling constants g and 5 can make the GES feature of (5) much richer in comparison with the Calogero model. For example, while bosonic (fermionic) excitations in 28
Fractional Statistics
in Some Exactly Solvable Models
1881
the Calogero model occur only in the absence of long-range interaction, the quasiexcitations in (5) can behave as pure bosons (fermions) even in the presence of both the long-range interactions satisfying the constraint v(5,g) = 0 (u(5,g) = 1). Both of these constraints lead to the same parabolic curve g = —6(1 + 5). A family of such parabolas with shifted apex points are generated for other values of v and the lowest apex point is attained at v = i , where the quasi-excitations would behave as semions. 4. Conclusion Here we construct a many-particle quantum system (5) by adding some nonhermitian but combined parity and time reversal (PT) invariant interaction to the A^-i Calogero model. By using appropriate similarity transformations, we are able to map this extended Calogero model to a set of free harmonic oscillators and solve this model exactly. It turns out that this many-particle system with nonhermitian Hamiltonian yields a real spectrum. This fact supports the conjecture that the condition of hermiticity on a Hamiltonian can be replaced by the weaker condition of PT symmetry to ensure that the corresponding eigenvalues would be real ones. It is also found that the spectrum of extended Calogero model obeys a selection rule which leads to generalised exclusion statistics (GES). However, this extended Calogero model exhibits some remarkable properties which are absent in the case of usual Calogero model. For example, we curiously find that the GES parameter for this extended Calogero model differs from the corresponding exchange statistics parameter. Moreover a 'reflection symmetry' of the exchange statistics parameter, which is known to exist for A^-i Calogero model, can be found in the case of extended model only if the strength of P T invariant interaction exceeds a critical value. Finally we note that, it is possible to obtain another exactly solvable manyparticle quantum system by adding some nonhermitian but P T invariant interactions to the BN Calogero model (associated with BN Lie algebra). 12 Such a P T invariant model also leads to real spectrum with properties quite similar to the case of extended A^-i Calogero model. Acknowledgments I would like to thank Prof. A. Kundu and Dr. B.P. Mandal for collaborating with me in Ref. 7 and Ref. 12. I also like to thank Prof. M.L. Ge for kindly inviting me to the 'APCTP-Nankai Symposium', October 7-10, 2001, Tianjin, China. References 1. F. D. M. Haldane, Phys. Rev. Lett. 67, 937 (1991). 2. F. Calogero, J. Math. Phys. 10, 2191 (1969); J. Math. Phys. 12, 419 (1971). 3. B. Sutherland, J. Math. Phys. 12, 246 (1971). 29
1882 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
B. Basu-Mallick S.B. Isakov, Mod. Phys. Lett.B 8, 319 (1994); Int. J. Mod. Phys.A 9, 2563 (1994). M.V.N. Murthy and R. Shankar, Phys. Rev. Lett. 73, 3331 (1994). A.P. Polychronakos, in Les Houches Lectures (Summer 1998), hep-th/9902157. B. Basu-Mallick and A. Kundu, Phys. Rev. B 62, 9927 (2000). C M . Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); J. Phys. A 31, L273 (1998). C M . Bender, S. Boettcher and P.N. Meisinger, J. Math. Phys. 40, 2210 (1999). F.M. Fernandez, R. Guardiola, J. Ros and M. Znojil, J. Phys. A 32, 3105 (1999). A. Khare and B. P. Mandal, Phys. Lett. A 272, 53 (2000). B. Basu-Mallick and B.P. Mandal, Phys. Lett. A 284, 231 (2001). N. Gurappa and P.K. Panigrahi, Phys. Rev. B 59, R2490 (1999). K. Sogo, J. Phys. Soc. Jpn. 65, 3097 (1996). T.H. Baker and P.J. Forrester, Nucl. Phys. B 492, 682 (1997). H. Ujino, A. Nishino and M. Wadati, J. Phys. Soc. Jpn. 67, 2658 (1998).
30
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1883-1889 © World Scientific Publishing Company
THE ROTOR MODEL A N D COMBINATORICS
M. T. BATCHELOR and J. de GIER Centre for Mathematics and its Applications, and Department of Theoretical Physics, I.A.S., Australian National University, Canberra ACT 0200, Australia B. NIENHUIS Instituut voor Theoretische Fysica, Universiteit van 1018XE Amsterdam, The Netherlands
Amsterdam,
Received March 26, 2002 We examine the groundstate wavefunction of the rotor model for different boundary conditions. Three conjectures are made on the appearance of numbers enumerating alternating sign matrices. In addition to those occurring in the 0 ( n = 1) model we find the number Av(2m + 1;3), which 3-enumerates vertically symmetric alternating sign matrices.
1. Introduction The XXZ Heisenberg spin chain and the related six-vertex model stand as central pillars in the study of exactly solved models in statistical mechanics. 1 ' 2 It has been known for many years that, with appropriate boundary conditions, their groundstate energy is trivial at the particular anisotropy value A = —1/2. Only recently has it been realised that the corresponding groundstate wavefunction possesses some rather remarkable properties. 3-5 These observations extend to the related O(n) loop model 6 - 7 at n = l. 4 ' 8 " 10 Consider first the periodic antiferromagnetic XXZ chain
H = -\ £ (°M + i + * K + i + A O K H ) ,
(1)
defined on an odd number N of sites. Here (
1=0
x
31
Jl
1884 M. T. Batchelor, J. de Gier & B. Nienhuis
for size N = 2m +1. The remarkable point being that A(m) is the number o f m x m alternating sign matrices. 11 The resulting sequence A(m) = 1,2,7,42,429,7436... is also known to count other combinatorial objects. 12 ' 13 Moreover, these numbers appear in the sum of all the groundstate wavefunction components. These observations remain to be proved. An even number of sites and other boundary conditions have also been considered, both for the XXZ chain (twisted and closed8, quantum symmetric bc's) and the 0 ( n = 1) loop model (periodic and closed bc's). These see the appearance of other well known numbers counting alternating sign matrices and related objects in different symmetry classes. For example, with the smallest component of the groundstate wavefunction again unity, the 0 ( n = 1) loop model with closed boundary conditions has largest component given by Ay (2m — 1) for N = 2m — 1 and 7V8 (2m) for N = 2m. Here
is the number of (2m +1) x (2m + 1 ) vertically symmetric alternating sign matrices and
is the number of cyclically symmetric transpose complement plane partitions. The number JV8(2m) is conjectured to be AvH(4m+l)/Av(2m + l ) , where Ayn(4m+1) is the number of (4m + 1) x (4m + 1) vertically and horizontally symmetric alternating sign matrices. 1 4 ' 1 5 Another quantity, which appears for periodic boundary conditions, is A H T(2m) = A ( m )
2
n£rT,
<5>
the number of 2m x 2m half turn symmetric alternating sign matrices. Further developments include the combinatorial interpretation of the elements of the 0 ( n = 1) loop model wavefunction in terms of link patterns 8 - 1 0 and the relation to a one-dimensional stochastic process. 10 There has been some progress attempting to prove these conjectures using Bethe Ansatz techniques. 16 ' 17 In this paper, we examine the groundstate wavefunction of the rotor model 18 discussed by Martins and Nienhuis. The rotor model is based on a variant of the Temperley-Lieb algebra, which underpins the six-vertex model, the O(n) model and the critical Q-state Potts model. 1 ' 2 ' 1 9 The rotor model is denned in Section 2, with our results presented in Section 3. a
The standard nomenclature for these bc's is open be, but since these bc's are spin-conserving in the XXZ chain or loop reflecting in the 0 ( n = 1) model we find the term closed be more appropriate, here reserving open be for non-conserving boundary conditions. 32
The Rotor Model and Combinatorics
1885
Here we see the appearance of another number, Ay (2m + 1; 3), which is the 3enumeration of (2m + 1 ) x (2m +1) vertically symmetric alternating sign matrices, or equivalently, the number of vertically symmetric 6-vertex configurations with domain wall boundary conditions and A = - 1 / 2 . It is given by A v ( 2 m + 1;3) =
\{
2m
=
j(2j-1)P
L 5 ' ^ , 16038, • • -
(6)
In general, the ^-enumeration of alternating sign matrices in the terminology of Kuperberg , 15 is equivalent to the enumeration of six-vertex configurations with domain wall boundaries with A = 1 — x/2 and at the symmetric point with respect to the spectral parameter. We give some concluding remarks in Section 4. 2. T h e rotor model We suggest that the remarkable observations of this 0(n = 1) model are related to the combination of two key properties, namely solvability and the absence of finite size corrections to the groundstate energy. Now the 0(n = 1) model is not unique in this combination. Recently Martins and Nienhuis 18 introduced a model that shares the same two properties. In this so-called rotor model a set of loops covers all the edges of the square lattice precisely twice. At the vertices all the loops make a turn of 7r/2 which permits four types of vertices as displayed in Figure 1.
+
J L J _ _JL
R
Fig. 1.
L
A
D
Vertices of the rotor model.
A natural interpretation is that the loops are trajectories of particles, and that the two loop segments visiting the same edge are traversed in opposite directions. Thus the four kinds of vertices shown in Figure 1 behave as scatterers: right (R) and left (L) rotors, at which the particles always turn right and left respectively, and ascending (A) and descending (D) diagonal mirrors at which the particles get reflected. To clearly display the scatterers we propose that the particles always follow the left hand side of the road, as is customary in Australia where this paper was conceived. In a different interpretation the two loop segments at the same edge are the trajectories of different kinds of particles, traversed in either direction. Then the scatterers can all be interpreted as double mirrors on each site, each reflecting one kind of particle and transmitting the other. At the R and L sites these mirrors 33
1886
M. T. Batchelor, J. de Gier & B. Nienhuis
are placed crosswise, AD and DA respectively, while at the original ascending and descending mirrors, the double mirrors are placed parallel, AA and DD respectively. This alternate interpretation will not affect the distributions of trajectories in an infinite system, but it will result in changes on some finite systems. Martins and Nienhuis solved this model by means of the Yang-Baxter equation when these scatterers occur with the respective weights WR = WL = sin u cos(27r/3 — u), UA = sin(7r/3 — u) cos(27r/3 — u), WD = — sinucos(7r/3 — u).
(7)
independently at each vertex. In this paper we consider this model with periodic boundary conditions (pbc) and with closed boundaries at which the trajectories are reflected. We will be interested in the structure of the groundstate eigenvector. Since the transfer matrix as a function of u forms a commuting family, the groundstate is independent of u. Then it is convenient to consider the Hamiltonian, found (up to a constant) as the logarithmic derivative of the transfer matrix with respect to u at u = 0:
H = Y,3-Ri-Li~Ei-
(8)
i
For system size N the operators R, L and E are shown in terms of the loops in Figure 2.
H IX R
L Fig. 2.
E Generators.
Martins and Nienhuis showed that the operators L^i and i?2i-i generate a Temperley-Lieb (TL) algebra, and so do the operators -£21-1 and R^i- In periodic systems of even size, and in bounded systems these two TL algebras commute with each other. What changes the physics is the presence in the Hamiltonian of the term Ei — RiLi. Also the E% by themselves generate a TL algebra. In odd, periodic systems the odd and even sites cannot be distinguished. In this case the L and the R together form a TL algebra of 27V sites. When the system is odd and periodic, the interpretation of the R and L vertices as rotors or alternatively as crossing mirrors, will naturally result in different pbc. The rotor interpretation permits closed trajectories that wind the cylinder twice. In the alternative interpretation no closed winding trajectories are possible, and the odd system must have two unmatched terminals. In this paper we follow the latter interpretation. 34
The Rotor Model and Combinatorics 1887
The states of the model are the pairings of those terminals that are connected by a trajectory in the 'past' half of the strip or cylinder. When the system is periodic, one may distinguish the side of the cylinder along which the trajectory runs: a connection between site 1 and site N may pass all sites 2,...N—1, or it may simply connect site N to site N + 1 which is identified to 1. These two states can be distinguished, in which case we speak of pbc per se, or they may be identified, for which we reserve the phrase pbc with identified connectivities. 3. Results for the groundstate wavefunction The groundstate wavefunction of the Hamiltonian (8) satisfies the eigenvalue equation Hipo = 0. In this section we formulate three conjectures regarding V'o for the different types of boundary conditions discussed in Section 2. Conjecture 1: For closed boundary conditions, if the smallest element of the rotor model groundstate wavefunction for N — 1m — 1 is normalized to Ay (2m — 1;3), then all of its elements are integers and the sum of its elements is given by S(2m~l) = 3( m _ 1 ) JV8(2m). For N = 2m, normalize the groundstate wavefunction to the smallest integer such that all elements are integers, the sum of the elements is given by S(2m) = 32e™Aw(2m + l), where6m = 0 , 1 , 3 , 6 , 9 = [ ( m - l ) ( m + 2)/3J for m = 1 , . . . , 5 . This conjecture is based on the results presented in Table 1 and was checked up to N = 10. Conjecture 2: For periodic boundary conditions, normalize the smallest element of the rotor model groundstate wavefunction to the smallest integer such that all elements are integer. The sum of its elements is then given by S(2m — 1) = 3 3 r Mv(2ra + 1;3)2 for odd system sizes and by S(2m) = 3 m2 i4 H T(2m) for even system sizes. This conjecture is based on the results presented in Table 2 and was checked up to N = 9. Conjecture 3: For periodic boundary conditions and identified connectivities, normalize the smallest element of the rotor model groundstate wavefunction to the smallest integer such that all elements are integer. The sum of its elements is then given by S{2m) = 2e™A(m), where 6m = 0,1,3, 6,9,13 = [(m - l)(m + 2)/3j for TO = 1 , . . . , 6 . This conjecture is based on the results presented in Table 3 and was checked up to N = 12. 4. Discussion In this paper we have examined the groundstate wavefunction of the rotor model for three different boundary conditions. As for the 0 ( n = 1) model, numbers known to 35
1888
M. T. Batchelor, J. de Gier & B.
Nienhuis
Table 1. Groundstate wavefunctions of the rotor model with closed boundaries. Note t h a t by V>o = (2,1) with multiplicity (2, 2) we mean Vo = (2, 2 , 1 , 1 ) . N
m
ipo
multiplicity
1 2 3 4 5 6
1 1 2 2 3 3
(1) (1)
Sj^
1 1 6 (2,2) 27 (1,1,2) (2, 1, 4, 2, 4, 2, 4, 4, 2) 891 (1) (1)
(2,1) (14,5,4) (113, 111, 55, 31, 25, 21, 19, 11, 5) (4760, 1440, 1192, 1028, 601, 565, 326, 310, 126, 121, 86)
(1, 2, 4, 1, 4, 2, 2, 2, 1, 18954 2,4)
Table 2.
Groundstate wavefunctions of t h e rotor model with periodic boundaries.
N
m
T/JQ
multiplicity
1 2 3 4 5
1 1 2 2 3
(1)
(1)
SN'
1 6 27 810
(2,2) (2,1) (3,6) (5,2) (118, 35, 25, 22, 20, 5, 4) (2, 2, 8, 4, 8, 8, 4) (1036, 463, 208, 143, 127, 122, 65, 22, (5, 10, 10, 20, 5, 10, 20,18225 10) 10, 10)
Table 3. Groundstate wavefunctions of the rotor model with periodic boundaries and identified connectivities. N
m
ipo
multiplicity
S^N'
2 4 6 8
1 2 3 4
(1)
(1)
(2,1) (26, 9, 7, 2) (1798, 486, 410, 267, 234, 232, 165, 106, 90, 81, 76, 70, 56, 45, 20, 9, 4)
1 6 189
(2,2) (2, 3, 14, 6) (2, 8, 16, 2, 16, 16, 8, 30618 16, 4, 16, 8, 8, 16, 32, 16, 8, 4)
enumerate equally weighted alternating sign matrices appear in the normalization of the wavefunction. For the rotor model we also see the number Ay (2m + 1; 3), enumerating alternating sign matrices in which the minus signs have weight 3. 1 5 We find it quite surprising that the conjectures in Section 3 can be formulated at all. They are a result of the normalizations factoring into relatively small primes and thus enabling their recognition. This property appears to be absent for other boundary conditions, for example, pbc in the rotor interpretation for odd system sizes. It is even more remarkable that these numbers have a well known combinatorial meaning. Acknowledgements It is a great pleasure to dedicate this paper to Fred Wu on the occasion of his 70th birthday. This work has been supported by the Australian Research Council and the Dutch foundation 'Fundamenteel Onderzoek der Materie' (FOM). 36
The Rotor Model and Combinatorics 1889
References 1. E. H. Lieb and F. Y. Wu, in Phase Transitions and Critical Phenomena, Vol. 1, eds. C. Domb and M. S. Green (Academic Press, London, 1972) p. 331. 2. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982). 3. A. V. Razumov and Yu. G. Stroganov, J. Phys. A 34, 3185 (2001). 4. M. T. Batchelor, J. de Gier and B. Nienhuis, J. Phys. A 34, L265 (2001). 5. A. V. Razumov and Yu. G. Stroganov, J. Phys. A 34, 5335 (2001). 6. R. J. Baxter, S. B. Kelland and F. Y. Wu, J. Phys. A 9, 397 (1976). 7. H. W. J. Blote and B. Nienhuis, J. Phys. A 22, 1415 (1989). 8. A. V. Razumov and Yu. G. Stroganov, "Combinatorial nature of ground state vector of O(l) loop model", arXiv.math.CO/0104216. 9. A. V. Razumov and Yu. G. Stroganov, "O(l) loop model with different boundary conditions and symmetry classes of alternating-sign matrices", arXivxond-mat/0108103. 10. P. A. Pearce, V. Rittenberg and J. de Gier, "Critical Q = l Potts model and TemperleyLieb stochastic processes", arXiv:cond-mat/0108051. 11. W.H. Mills, D.P. Robbins and H. Rumsey, J. Combin. Theory A 34, 340 (1983). 12. D. M. Bressoud, Proofs and Confirmations—The story of the alternating sign matrix conjecture (Cambridge University Press, 1999). 13. J. Propp, "The many faces of alternating sign matrices", preprint. 14. D. P. Robbins, "Symmetry classes of alternating sign matrices", arXiv:math.CO/0008045. 15. G. Kuperberg, "Symmetry classes of alternating-sign matrices under one roof", arXiv:math.CO/0008184. 16. J. de Gier, M. T. Batchelor, B. Nienhuis and S. Mitra, "The XXZ spin chain at A = —5: Elementary symmetric functions, Schur functions and determinants", arXiv:math-ph/0110011. 17. N. Kitanine, J. M. Maillet, N. A. Slavnov and V. Terras, "Emptiness formation probability of the XXZ s p i n - j Heisenberg chain at A = 3 " , arXiv:hep-th/0201134. 18. M. J. Martins and B. Nienhuis, J. Phys. A 3 1 , L723 (1998). 19. F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982).
37
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1891-1897 © World Scientific Publishing Company
P H A S E TRANSITIONS IN T H E T W O - D I M E N S I O N A L 0 ( 3 ) MODEL
HENK W.J. B L O T E Laboratory of Applied Physics, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands and Lorentz Institute for Theoretical Physics, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands WENAN GUO Physics Department, Beijing Normal University Beijing 100875, P. R. China
Received 9 November 2001 We use Monte Carlo, transfer-matrix and finite-size scaling methods to investigate twodimensional O(n) models with n > 2, in particular t h e case n = 3 which includes the classical Heisenberg model. Depending on the t y p e of interaction and the lattice structure, two different types of phase transitions are present. One type resembles t h e hard-hexagon transition and occurs in the loop representation of the honeycomb O(n) model. T h e other type is a first-order transition which occurs for spin-spin interactions that are strongly nonlinear in the neighbor-spin products. When the nonlinearity is decreased, t h e first-order line ends in a critical point. T h e existence of the first-order line is in agreement with mean-field theory as well as with high- and low-temperature approximations.
1. I n t r o d u c t i o n The 0(n) spin model represents a system of interacting n-dimensional vectors s = ) on a lattice. The O(n) symmetry implies that the Hamiltonian is invariant under rotations in the space of the spin vectors. We consider the case of 0(n) symmetric pair interactions
H/kBT = -J2
h
(si-Sj)
(1)
where the sum is on all pairs of nearest neighbors, and h is an arbitrary function with implicit temperature dependence. The 'linear' case, i.e. h(x) = Kx where K is the coupling constant, includes the classical XY model (n = 2) and the classical Heisenberg model (n = 3). For reasonable choices of the function h, in particular monotonically increasing functions, it is plausible that the model belongs to the same universality class as the linear model. 39
1892
H. W. J. Blote & W. Guo
The question whether or not two-dimensional O(n) models with n > 2, and in particular with n = 3, undergo a phase transition at a sufficiently low temperature has received considerable attention; 1 " 6 see also references therein. These papers contain conflicting answers. However, according to the prevailing interpretation, ordering transitions are absent in the O(n) model with n > 2. The main argument relies on the spin-wave result 1 ' 3 by Bloch that a spontaneously magnetized state cannot exist at a nonzero temperature. This does not qualify as a proof for the absence of a transition: the spin-wave argument applies as well to the XY (n = 2) model where a phase transition 7 is known to occur (but not to a spontaneously magnetized state). The latter transition is however linked to topological excitations (vortices) which lack relevance for n — 3. This, together with an exact result of Kunz and Wu 4 which excludes phase transitions in a part of the n > 2 parameter space, forms the basis of the above-mentioned prevailing interpretation. However, here we describe two sorts of transition for n > 2. The first type, which occurs in the O(n) loop model on the honeycomb lattice, is described in section 2. It is unphysical in the spin representation, because negative Boltzmann weights occur. It is thus consistent with the hypothesis that phase transitions are absent in two-dimensional spin models with n > 2. However, in section 3 we describe a phase transition in a genuine Heisenberg-type 0(3) model. The phase transition does not lead to a long-range ordered state (in the sense of a nonzero spontaneous magnetization) and is therefore consistent with the spin-wave theory.
2. P h a s e Transition in t h e Loop Model The 'loop' version of the 0 ( n ) model is defined by the choice h(x) = log(l + ax), where the parameter a is an inverse-temperature-like parameter, and the normalization Si.Si = n. The model on the honeycomb lattice can be mapped 5 onto a gas of nonintersecting loops running over the edges of the honeycomb lattice. Each edge covered by a loop carries a Boltzmann weight a, and each loop a weight n. The loop representation has enabled exact solutions along special lines in the n, a plane for n < 2 8_11 and for n > 2. 12 From these solutions we know that for n < 2 an ordering transition occurs at finite values of x, and that for n > 2 the loop model is in a long-range ordered state for x = oo. This state is not of the 'ferromagnetic' type, but chooses between 3 sublattices, and reminds of the hard-hexagon model. In the absence of exact solutions in most of the n, x plane, we have applied Monte Carlo and transfer-matrix techniques. 13 Surprisingly, a transition to the long-range ordered state was found at finite values x < oo. The hard-hexagon-like critical line spans the range 2 < n < oo. The resulting phase diagram is shown in fig. 1. Also shown are the boundary of the 'physical region' of the spin model, where the energy h(si.Sj) is real for all Si.Sj (curved line in the middle), and a region where a transition is rigorously excluded (above the curved line on the far right). The latter line is based on the work by Kunz and Wu;4 see also an erratum. 13 We observe that the newly found critical line indeed avoids the excluded region. 40
Phase Transitions
in the Two-Dimensional
0(3) Model
1893
4 3.5 3 2.5
I
2 1.5 1 0.5 5
0
10
20
40 80 infty
n
Fig. 1. Phase diagram of the 0 ( n ) model on the honeycomb lattice in the n-a plane. The horizontal scale is chosen as 1 — 8/(n -f 10) in order to show the whole range up to n = oo. T h e vertical scale displays W(n) = l/[(n + 10)l/6a]. The data points show our results for the newly found phase transition. The curve in the range — 2 < n < 2 shows an exact solution. The region on t h e right hand side of curve ending at n = W = 0 is unphysical in the spin representation. Rigorous arguments exclude phase transitions in the region indicated at the upper right.
It is completely embedded in the unphysical region of the spin model, but it is physical in the language of the O(n) loop model. 3. The Strongly Nonlinear 0 ( 3 ) Model We use Eq. (1) for the Heisenberg case n = 3 with a spin-spin interaction h{si • S^ = 2K[{1 + ^ • Sj)/2}p
(2)
with spins normalized to length 1. The parameter p determines the degree of nonlinearity of the energy function h. This form is chosen as to avoid powers of negative numbers, and to limit the energy range to 2K, as in the linear case. 3 . 1 . Mean-field
theory
Consider a spin si, interacting with z neighbors. Denote the average magnetization of the spin system as m, say along the x-axis. The local energy is Eioc = -2zK[(l
+ ^ • m ) / 2 f = -2zK[(l
+ xm)/2}p
(3)
and the thermal average of the x-component of s~l satisfies (x) = /
xe~Bl°"dx/
J
e~E^dx
(4)
For large enough K, the self-consistency equation (a;) = m has solutions at nonzero m. While m is a decreasing function of K, this function depends qualitatively on 41
1894
H. W. J. Blote & W. Guo
p. For small enough values of p, m decreases continuously to 0 at the critical point K = Kc. We thus find Kc by solving d{x)/dm = 1 which leads to Kc = 2P- 3 3/p
(5)
For larger values p, (x) increases faster than linear with m for small m, but still levels off at large m. The nozero solutions of (x) = m thus become duplicate, i.e., the transition turns first order for large p. These two distinct ranges of p are separated by the tricritical point, which can be determined by solving for K and p in d(x)/dm = 1 and d3(x)/dm? = 0 at m = 0. The second equation expresses the absence of the lowest order of nonlinearity of (a;) as a function of m. The tricritical coordinates are ptli = (—1 + >/33)/2 and KtT1 = 2 P t r i ~ 3 3/p t r i . For p > ptri, the first-order transition point follows by equating the areas enclosed by the (x) vs. m curve and the (x) = m line. Thus we have determined the first order line numerically; the resulting phase diagram is shown in fig. 2.
3 2.5 2 1.5 1 0.5 J
0 0
1
I
2
I
3
I
4
1
5
I
6
L
7
8
P
Fig. 2. Mean-field phase diagram of the nonlinear 0 ( 3 ) model on the square lattice, in the K versus p plane. The line of phase transitions consists of two parts: a continuous transition at small p, and a first-order part. T h e two parts are separated by a tricritical point (asterisk).
3.2. High' and low-temperature
approximations
Estimates of the location of a first-order transition(if any) can also be obtained from intersections of high- and low-temperature approximations of the free energy. Neglecting loop diagrams in the high-temperature expansion the lattice effectively reduces to the Bethe lattice, for which we obtain the free energy via a transfermatrix-like approach. The partition function Z\, 'per bond' between spins s and t equals Zh=
I dsexp{2£T[(l + s i ) / 2 ] p } 42
(6)
Phase Transitions in the Two-Dimensional 0(3) Model 1895
which is independent of t. Expansion of the exponential function yields oo
where the prefactor accounts for the spin degrees of freedom and the sum for the spin-spin interaction. The partition function of a Bethe lattice with N spins, each interacting with z neighbors, and zN/2 bonds follows as
For z = A, the high-temperature approximation of the free energy is thus
II--'^Ht
(9)
We use a low temperature approximation for spins almost aligned along the z axis: Si = I sf,s%, -i/l — sf2 — S? ) where the x and y components are small. Small deviations between neighbors i and j increase the energy per bond Ev/kT + 2K= ±PK[(s* - s*f + (a? - sf)2]
(10)
i.e. the Gaussian model applies to this quadratic form. After a Fourier transformation it is straightforward to obtain the partition function; the free energy follows as NkT -4K
N
- log(47r) + log(8PK) + 1 Y,
lo
S[(sin \**)2 + (sin \kyf)
(11)
k lo
For large N the sum satisfies ^ Y.k g [ ( \ ^)2 + ( s i n \ky)2\ - -0.2200507 • • •. The low- and high-temperature approximations of the free energy are found to intersect, and thus predict the approximate location of a possible first-order transition line. These intersections were found numerically, and shown in fig. 3. 3.3. Monte-Carlo
sin k
results
The model defined by Eqs. (1) and (2) was investigated by a conventional Monte Carlo algorithm with local spin updates. Randomly chosen orientations for the spin vectors were accepted or rejected with Metropolis-type probabilities. The autocorrelation times are found to increase considerably at low temperatures, especially for system sizes L exceeding about 100. The efficiency of the algorithm decreases even further at high values of p where the acceptance ratio becomes small. Nevertheless we could resolve the phase diagram. No signs of a phase transition were found for p = 1, i.e. the linear Heisenberg model. But for larger p, pronounced 43
1896
H. W. J. Blote & W. Guo
1.8 1.75 1.7
1.65 1.6
*
1.55
1.5 1.45 1.4 1.35 1.3
10
20
30
40
50
60
p
Fig. 3. Monte Carlo results for t h e phase diagram of the two-dimensional 0 ( n ) spin model on the square lattice, in the K versus p plane. Results from the intersections of low- and high-temperature approximations of the free energy are included as well (full curve).
maxima in the heat capacity appear, and for p > 16, we find numerical evidence for a divergence of the heat capacity. For p > 18, the simulations reveal a jump in the energy as a function of K, and a clear hysteresis effect. The first-order character becomes even stronger at larger p. The transition for p > 20 was found by Monte Carlo runs starting from a spin configuration of which one half was fully aligned, and the other half filled with randomly chosen spins. For p < 20 we determined the location Kmax(p, L) of the heat capacity maximum c max (p, L) as a function of K for system sizes up to L = 48, and extrapolated to L = oo. The heat capacity does not seem to diverge for p < 16. For p = 16 we observe a divergence approximately as L 7 / 4 , which indicates the presence of an Isinglike critical point. For p > 16 the divergence agrees with first-order behavior cmax(p, L) oc L2. The Monte Carlo data are included in fig. 3. 4. Discussion We have provided evidence for two types of phase transitions in O(n) models with n > 2. The first type is unphysical in the spin language, and depends essentially on the underlying lattice structure. The second type is, however, found in a pure spin system, in a conspicuous disagreement with expectations formulated in the literature. We review the evidence presented above. In low-dimensional models, mean-field theory tends to predict continuous phase transitions where they do not exist. The example found in section 3.1 should thus not be taken too seriously. The predicted first-order transition is more credible, because the strength of the predicted discontinuity increases at large p, and the role of fluctuations may thus be reduced. Another defect of mean-field theory is that it is based on an order parameter m that is actually zero. 1 ' 3 However, the 44
Phase Transitions in the Two-Dimensional 0(3) Model 1897 long-wavelength spin waves, which are responsible for t h e suppression of m , hardly affect correlations at distances of a few lattice units. T h e mean-field result can thus still be r e g a r d e d as suggestive of a first-order transition at large p. Likewise, t h e result of the high- and low-temperature approximations is less t h a n compelling. Nevertheless, here also the predicted energy j u m p increases with p: t h e two free-energy branches are pushed far away, mimicking high- and lowt e m p e r a t u r e configurations with only limited fluctuations. This lends some support t o our approximation. T h e resulting first-order line is in a reasonable qualitative agreement w i t h Monte Carlo results (fig. 3). It tends t o become better at large p. T h e Monte C a r l o runs provided a clear first-order picture; t h e numerical errors are very small in comparision to the differences with t h e two analytic approximations for t h e first-order line. T h e Monte Carlo results are clearly superior. Finally we mention t h a t similar transitions may occur for larger values of n , and t h a t Monte Carlo results of D o m a n y et al. 1 4 for the analogous strongly nonlinear 0 ( 2 ) model showed t h a t t h e Kosterlitz-Thouless transition is preempted by a first order one.
Acknowledgements It is a pleasure t o t h a n k Prof. H.J. Hilhorst, Prof. H. Kunz and Prof. F.Y. W u for valuable discussions. This research was supported in part by the Dutch F O M (Fundamenteel Onderzoek der Materie) foundation a n d by the National Science Foundation of C h i n a under Grant #10105001. H.B. gratefully acknowledges t h e hospitality of t h e Physics Department of the Beijing Normal University, where p a r t of this work w a s performed.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
F. Bloch, Z. Phys. 6 1 , 206 (1930). H.E. Stanley and T.A. Kaplan, Phys. Rev. Lett. 17, 913 (1966). N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966). H. Kunz and F.Y. Wu, J. Phys. A 21, L1141 (1988). E. Domany, D. Mukamel, B. Nienhuis and A. Schwimmer, Nucl. Phys. B 190 [FS3], 279 (1981). A. Patrascioiu and E. Seiler, Phys. Lett. B 430, 314 (1998). J. M. Kosterlitz and D.J. Thouless, J. Phys. (7 5, L124 (1972); J. Phys. C 6, 1181 (1973); J.M. Kosterlitz, J. Phys. C 7 , 1046 (1974). B. Nienhuis, Phys. Rev. Lett. 49, 1062 (1982). R.J. Baxter, J. Phys. A 19, 2821 (1986); J. Phys. A 20, 5241 (1987). M.T. Batchelor and H.W.J. Blote, Phys. Rev. Lett. 6 1 , 138 (1988); Phys. Rev. 5 39, 2391 (1989). J. Suzuki, J. Phys. Soc. Jpn. 57, 2966 (1988). R.J. Baxter, J. Math. Phys. 1 1 , 784 (1970). W.-A. Guo, H.W.J. Blote and F.Y. Wu, Phys. Rev. Lett. 85, 3874 (2000). E. Domany, M. Schick and R.H. Swendsen, Phys. Rev. Lett. 52, 1535 (1984).
45
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1899-1905 © World Scientific Publishing Company
THE 8 V C S O S MODEL A N D T H E sl2 LOOP ALGEBRA S Y M M E T R Y OF T H E SIX-VERTEX MODEL AT ROOTS OF U N I T Y
TETSUO DEGUCHI * Department
of Physics, Ochanomizu University, 2-1-1 Ohtuska Bunkyo-ku, Tokyo 112-8610, Japan
Received 6 November 2001 We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group ET,ri(sl2) at the discrete coupling constants: 2Nr) = m i -\-imir, where N, mi and m^ are integers. Then we show that degenerate eigenvectors of the transfer matrix of t h e six-vertex model at roots of unity in the sector Sz = 0 (mod iV) are derived from those of t h e 8V CSOS model, through the trigonometric limit. They are associated with t h e complete N strings. Prom the result we see that the dimension of a given degenerate eigenspace in the sector Sz = 0 (mod N) of the six-vertex model at iVth roots of unity is given by 2 2 S n > ^ / J V , where S^ax is the maximal value of the total spin operator Sz in the degenerate eigenspace.
1. I n t r o d u c t i o n Recently, it has been explicitly discussed that the transfer matrix of the six-vertex model at roots of unity has the symmetry of the sh loop algebra.1^5 Let us consider the XXZ spin chain under the periodic boundary conditions L
HXxz = ~ J E ipfCTf+i+ aJaJ+i + A(Tf a!+i) •
(^
Here the parameter A is related to the q variable of the quantum group Uq(sl2) as
A = i(? + g- 1 ). 2N
1
(2)
When q = 1, it was shown that the XXZ Hamiltonian commutes with the generators of the sl2 loop algebra, which is an infinite dimensional algebra. Furthermore, it was shown1 by the Jordan-Wigner method for N = 2 and numerically for general N that the dimensions of the degenerate eigenvectors are given by some powers of 2, which increase exponentially with respect to the system size L. * [email protected] 47
1900
T.
Deguchi
The exponential degeneracy of the sl2 loop algebra should be important for the problem of the "completeness of the Bethe ansatz eigenvectors". In fact, the si2 loop algebra symmetry has not been considered in the standard arguments of the string hypothesis. 6 ' 7 Thus, it seems that it is still open whether we can construct 2L linearly independent eigenvectors of the XXZ spin chain at roots of unity for general L. The question should be related to so called singular Bethe ansatz solutions. 8 In fact, it is numerically confirmed that the standard solutions of the Bethe ansatz equations determine only eigenvectors which have the highest weights of the sl2 loop algebra. 2 Furthermore, some important properties of complete N strings have been discussed in association with the sl2 loop algebra.2"4 Interestingly, it was numerically suggested that the transfer matrix of the eightvertex model at the discrete coupling parameters should have the degenerate eigenvectors corresponding to the degeneracy of the sl2 loop algebra. 1 Furthermore, it has been recently shown that some degenerate eigenspace of the eight-vertex model has dimension of N2LIN if L/N is an even integer.5 Let us consider the XYZ Hamiltonian under the periodic boundary conditions 9 ' 10 L
HXYZ
— - 2 J {Jxvfaf+i
where the coupling constants Jx, Jy Jx = J(l + ksn2(2ri)),
+ JYOJ(JJ+1 an
+ Jzafaf+1)
,
(3)
d Jz are given by
JY = J ( l - ksn2(2ri)),
Jz = Jcn(2j])dn(2r)).
(4)
Here sn(z), cn(z) and dn(z) denote the Jacobian elliptic functions with elliptic modulus A;. We have called 2r) the coupling parameter of the model. The number N has been related to 2r] by 2Nr) = 2m\K + im2K'. The symbols K and K' denote the complete elliptic integrals of the first and second kinds, respectively. In this paper, we discuss an algebraic construction of degenerate eigenvectors of the eight-vertex cyclic Solid-on-Solid model 11 " 13 (8V CSOS model), which is a variant of the eight-vertex Restricted Solid-on-Solid model (ABF model) Then, we show that through some limit, they give the degenerate eigenvectors of the sixvertex model in the sector Sz = 0 (mod N) consisting of the complete N strings. 2. The SI2 loop algebra symmetry of t h e XXZ spin chain Let us consider representations of the generators of Uq{sl2) on the £ t h tensor product of spin 1/2 representations. / = / /
8
® - ® / /
J
(5)
s * = x > f = x y */2 ® • • •
J=I
Let us introduce some symbols: [n] = (qn — q~n)/(q - q~l) for n > 0 and [0] = 1;
W! = nL:W-Setting 5 ± W = lim^^1(5±)iv/[iV]! 48
(7)
(6)
The 8V CSOS Model and the sfo Loop Algebra Symmetry
the operators S^N^
1901
are non-vanishing and we have
n+f/Vl
V^
5
2^
H-rrz
2-nz
9
®---®q~
+
(M-2)
z
® er* ® 9 ^ ^
(N-2)
z
® • • • g ^ ^
l<jl<-<JAT
(JV-4) _ Z
®
i
N „Z
(8) • • • ® a * , g ~
N „Z
2 CT
® • • • g
,
T
^
.
.
(8)
The study of the symmetries of the XXZ Hamiltonian under periodic boundary conditions at roots of unity was initiated in Ref.:15 S±(-N^ commute with the Hamiltonian (1) when Sz/N is an integer and q2N = 1 holds. However, there exists a much larger symmetry algebra than that of S^^.1 We remark that the XXZ Hamiltonian is associated with the affine quantum group Uq(sl2)- For instance, we may consider the following: T±
= E r * = E « " z / a ® • • -i'"2'2 ® at ® q"z/2 ® • • • ® q"z/2' J=I
(Q)
J=I
which is also obtained from 5 ± by the replacement q -» g _ 1 . When g2JV = 1, we define T±(^N^ similarly as in eq. (7). Let Tev(v) denotes the (inhomogeneous) transfer matrix of the six-vertex model. Then we can show the (anti) commutation relations when Sz = 0 (mod N) 1 = qNT6V(v)S^N\
S^Teviv)
= qNT6V(v)T±^
T ^ T ^ )
(10)
and therefore in the sector Sz = 0 (mod N) we have [S±w,H]
= [T±(N\H]=0.
(11)
Let us discuss the symmetry algebra. With the following identification1 N eo = 5+w, f0 = s-( \ e i = r - w , h=T+W, t0 = -t1 = -(-q)NSz/N,
(12)
we can show that they satisfy the defining relations of the s/2 loop algebra: [5+W.T+W] = [ S - W . T - W ] [S±(N);
SZ]
= ±NS±(N);
[T±(N)
^ gZ]
=
±NT±(N)
iSr+(JV)3j-(iV)
_ 35+(JV)2T-(iV)5,+(iV)
+35+(JV)r-(Ar)5+(iV)2
5 -(JV)3 r +(iV)
_ 2S-(N)2T+(N)S-(N)
+ 3S-(N)T+(N)S~(N)2
T+(N)35-(N) _
3T+(N)'2S-(N)T+(N)
T-(N)3g+(N)
3T-(^)25+(iV)T-(JV)
_
(13)
= 0) >
(14)
_T-(N)S+(N)3
=
_ T+(N)S-(N)3
=
Q^
=
Q^
=
Q)
+
3T+(N)g-(N)T+(N)2
_ g-(N)T+(N)3
+
3 T -(iV) 5 ,+(Ar) r -(JV)2 _ £ + ( ^ - ( ^ 3
Q^
(15) z
and in the sector S = 0(mod N) we have [ S + W ) 5 - W ] = [T+W,T"W] = - ( - g ) " - | s * .
(16)
The loop algebras with higher ranks are also discussed for some vertex models. 16 49
1902
T. Deguchi
3. T h e algebraic B e t h e ansatz of the elliptic quantum group The elliptic algebra ETtT](sl2) is an algebra generated by meromorphic functions of a variable h and the matrix elements of a matrix L(z, A) with non-commutative entries, 1 7 , 1 8 which satisfy the Yang-Baxter relation with a dynamical shift fl(12)(*i2, A - 2Vh^)L^{Zl,
\)L
= L^(z2,X)L^(z1,X-2r1h^)R^\z12,X)
.
(17)
Here h is a generator of the Cartan subalgebra h of sl2. Drinfeld's quasi-Hopf algebra gives a natural framework for the dynamical Yang-Baxter relation, which can be derived from the standard quantum group Uq(sl2) through the twist. 19 ' 20 The ^-matrix of (17) is essentially that of the ABF model14 (the 8V RSOS model). Let V be the two-dimensional complex vector space with the basis e[l] and e[— 1]. Here we denote e[—1] also as e[2], and let Eij denote the matrix satisfying Eije[k] = 6jke[i}. Then, the .R-matrix R(z, A) G End(V) is given by R(z, A; 77, T ) =
En En + E22 0 E22 + a{z, \)En E22
+(3(z, X)E12 ® E21 +/3(z, -X)E21
E12 + a(z, -X)E22 ® En ,
(18)
where h = En — E22 and a(z, A) and (3(z, A) are defined by a(z x)
^x)--e(z-2V)9(x)-
> -e(z-2V)e(x)>
(19)
The theta function has been given by oo
6(Z;T)
1/4
= 2p simrz
J J ( 1 -p2n)(l
-p2nexp(2mz)){l
-p2nexp(-2mz)),
(20)
n=l
where the nome p is related to the parameter r by p — exp(7rir) with Im r > 0 . Let us now review the construction of the eigenvectors of the elliptic algebra ETtn(sl2) at the discrete coupling parameter: 2JVT? = mi + m2T , where N, mi and 77i2 are any given integers. 5 Here we note that 2Nr) = mi + m2T corresponds to 27V77 = 277ii.ftT + im2K in (4). Hereafter we assume m 2 = 0 for simplicity. Let W = V(z\) ® • • • <8> V(ZL) be the Lth tensor product of the evaluation modules VA^ZJYS with Aj = 1 for all j . 1 7 ' 1 8 The transfer matrix T{z) of ET>n(sl2) is given by the trace of the L-operator acting on the module W L
L(z,X) = R(01\z
- ZI, A - 2 T ? ^ h{j)) L
x R^2\z - z2, X - 2r)Y,hU))
• • - i i ( 0 L ) ^ - zL,X)
(21)
Let us consider the mth product of the creation operators b(tj)'s on the vacuum. 1 8 ' 2 1 Let us assume the number m satisfies the following condition 2m = L-rN,
for r G Z 50
(22)
The 8V CSOS Model and the s/2 Loop Algebra Symmetry
1903
Hereafter we also assume that rm\ is even. We introduce a function gc(X) by <7C(A) = ecX Y[™=1 (&{X — 2rij)/6(2r])).
Vector vc is defined by vc = gc(X)v0,
where VQ is t h e
highest weight vector of W: hvo = Lv$. Then, making use of the fundamental commutation relations 18 associated with b(zj)'s, we can show that b(ti) • • • b(tm)vc is an eigenvector of the transfer matrix T(z) with the eigenvalue CQ(Z)
r 0[u„\j
- P-*vc n e(w ~ *>+27?) +e + P^c f r e(w - li -27?) rr /J
e(w-tj)
jl
9(w-tj)
9 w Za
( - ^
}Ae(v,-za-2n)' (23)
if rapidities t±, t2,..., tm satisfy the Bethe ansatz equations
The vector b(ti) • • • b(tm)vc is explicitly given by the following:18
( ^ ire , (J+ « x
11
/POP/,
11
s
E ft n ){T::B%
,(
_2
}
^-^10)
(25)
Here aj denotes the Pauli matrix a~ acting on the j t h site, <S the symmetric group, |0) the vacuum vector and fjk — 6{tj — ifc — 2rf)/6{tj — i*). 4. T h e eigenvectors of the 8V CSOS model Let us replace A with A + Ao in the L-operator (21) on W. Here Ao is independent of A. Then, the .R-matrix R(z,X + Ao) is related to the Boltzmann weights w(a, b, c, d; z, A0) of the 8V CSOS model through the following relation R(z,
—2r)d + Ao)e[c - d] <S> e[b - c] = ^
w(a, b, c, d; z, Xo)e[b — a]<8>e[a-d]
(26)
a
Here a, b, c, d denote the spin variables of the IRF (the Interaction Round a Face) model which take integer values.10 The spin variables have the constraint that the difference between the values of two nearest-neighboring spins should be given by ± 1 . Furthermore, for the 8V CSOS model discussed in Refs., 11-13 the spin variables take the restricted values such as 0, 1, . . . , N — 1 where the values 0 and N — 1 can be assigned for adjacent spins. Through the relation (26), we can show that the transfer matrix T(z) of ETtT){sl2) acting on the "path basis" corresponds to that of the 8V CSOS model. 10,18 Here we note that a "path" is given by a sequence of spin values satisfying the constraints on adjacent spins. Explicitly we consider the following18 \a\, a2, • • • CLL)W
= S(X + 27701) e[a\ - a2] e[a2 - a3] • • • e[a,L - ai] 51
(27)
1904
T. Deguchi
Here for the 8V CSOS model, we assume that az, — ai = ± 1 (mod N). Expressing the eigenvector b(t\) • • • b(tm)vc of T(z) in terms of the path basis, we obtain that of the transfer matrix of the 8V CSOS model. 5. The degenerate eigenvectors of t h e transfer matrix of t h e 8 V CSOS model Let us now assume that out of m rapidities ti,...,tm, the first R rapidities tj for j = 1,...,R are of standard ones satisfying the Bethe ansatz equations (24) with TO replaced by R, while the remaining NF rapidities are formal solutions given by Ha,j)=t{a)+i1(2j-N-l)+eT^),
for j = l,...,JV.
(28)
We call the set of N rapidities t(a,i)> • • • > t(a,N)i the complete iV-string with center i( Q ). Here the index a runs from 1 to F. Furthermore, we assume that the index (a, j) corresponds to the number R + N(a — 1) + j for 1 < a < F and 1 < j < N. We note that the complete strings were suggested in Ref.10 in another context. Using the fundamental commutation relations, we can show when £ ^ 0 T(z)b(h) • • • b(tR+NF)
(
R
vc = C 0 (z)&(ti) • • • b(tR+NF)
vc
R+NF\
E+
E
j=l
j^R+lJ
Cjb(t1)---b(tj-1)b(z)b(tj+1)---b(tR+NF)vc.
(29)
We divide eq. (29) by e, and send e to zero. Then, we can show that each of the terms of eq. (29) indeed converges, by making use of the following formula
11
fpaP0-
U
l
U x
l
11 [e(t l<j
x
V
•>
tkK_2f,))
'(30)
" '
for P £ Sm. Here H(x) denotes the Heaviside step function: H(x) = 1 for x > 0, H(x) = 0 otherwise. The symbol P S Sm denotes an element P of the symmetric group of m elements, where j is sent t o Pj G { 1 , 2 , . . . ,TO}for j — 1 , . . .TO.The formula (30) has been proven in Ref..22 Let us consider the following function of variable z5 rM
_V%-4,ca
{)=
h
FT
£0(z-tk+r,(2j-N+l))
M^'-tu+T&i-N-*)) J^e(z~Z0 + n(2j-N-3)) Uetz-zp + vW-N-i))
[6i)
Hereafter we assume exp(ANrjc) = 1. Then, the centers i( a ) 's are determined by G(z = t{a))=0,
for a = l,...,F.
(32)
We can show that the zeros of (32) also form complete iV strings, and also that the number of zeros of (32) is given by L — 2R, by using the Bethe ansatz equations 52
The 8V CSOS Model and the sfo Loop Algebra Symmetry
1905
(24). 5 T h u s , t h e n u m b e r of independent solutions to (32) is given by (L — 2R)/N, which leads t o t h e dimension 2^L~2IV>/N t h r o u g h the binomial expansion. T h u s , for the transfer m a t r i x of t h e 8V CSOS model, any standard Bethe ansatz eigenvector with R rapidities has t h e degeneracy of 2^L~2R^N. Let us now consider t h e connection of t h e CSOS model to the six-vertex model. Taking t h e trigonometric limit: r —> ioo a n d sending Ao to infinity with some gauge transformations, t h e L-operator of t h e 8V CSOS model becomes that of t h e sixvertex model. We m a y assume t h a t t h e trigonometric limits of the R rapidities of the B e t h e a n s a t z equations (24) w i t h exp(4?/c) = 1 satisfy the trigonometric Bethe ansatz e q u a t i o n s of t h e six-vertex model. Then, the degenerate eigenvectors with F complete N strings for t h e 8V C S O S model become those of the six-vertex model with F complete N strings. T h u s , we have shown t h a t the corresponding degenerate eigenspace is s p a n n e d by t h e eigenvectors having complete N strings, and also t h a t the dimension is given by 2^L~2IV>/N = 22S™°*/N since the highest weight Smax is given by L/2 — R. T h e result should b e consistent with the previous s t u d i e s . 1 - 4 Acknowledgements T h e a u t h o r would like t o t h a n k Prof. Y. Akutsu, Prof. K. Fabricius and Prof. B.M. McCoy for helpful discussions a n d valuable comments. He is thankful to Prof. M.L. Ge for his k i n d invitation t o t h e workshop "Nankai Symposium", October 8-11, 2001, Tianjin, China. This work is partially supported by the Grant-in-Aid for E n c o u r a g e m e n t of Young Scientists (No. 12740231). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
T. Deguchi, K. Fabricius and B.M. McCoy, J. Stat. Phys. 102, 701 (2001). K. Fabricius and B.M. McCoy, J. Stat. Phys. 103, 647 (2001). K. Fabricius and B.M. McCoy, J. Stat. Phys. 104, 575 (2001). K. Fabricius and B.M. McCoy, cond-mat/0108057. T. Deguchi, cond-mat/0109078 (submitted to J. Phys. A). M. Takahashi and M. Suzuki, Prog, of Theor. Phys. 46, 2187 (1972). A.N. Kirillov and N.A. Liskova, J. Phys. A 30, 1209 (1997). J.D. Noh, D.-S. Lee and D. Kim, Physica A 287, 167 (2000). Baxter, Ann. Phys. 70, 193 (1972). R. Baxter, Ann. Phys. 76, 1 (1973); 76, 25 (1973); 76, 48 (1973). P.A. Pearce and K.A. Seaton, Phys. Rev. Lett. 60, 1347 (1988). A. Kuniba and T. Yajima, J. Stat. Phys. 52, 829 (1987). Y. Akutsu, T. Deguchi and M. Wadati, J. Phys. Soc. Jpn. 57, 1173 (1988). G.E. Andrews, R.J. Baxter and P.J. Forrester, J. Stat. Phys. 35, 193 (1984). V. Pasquier and H. Saleur, Nucl. Phys. B330, 523 (1990). C. Korff and B.M. McCoy, hep-th/0104120. G. Felder and A. Varchenko, Commun. Math. Phys. 181(1996) 741 . G. Felder and A. Varchenko, Nucl. Phys. B 480 (1996) 485. O. Babelon, D. Bernard, E. Billey, Phys. Lett. B 375, 89 (1996). M. Jimbo, H. Konno, S. Odake, J. Shiraishi, Transformation Groups 4, 303 (1999). L. Takhtajan and L. Faddeev, Russ. Math. Survey 34(5), 11 (1979). T. Deguchi, cond-mat/0107260, to appear in J. Phys. A (2001). 53
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1907-1914 © World Scientific Publishing Company
T H E C H E R N - S I M O N S INVARIANT IN T H E B E R R Y PHASE OF A T W O B Y TWO HAMILTONIAN
DUNG-HAI LEE Department
of Physics,
University of California, Berkeley,
CA 94720
Received 26 October 2001 By varying (x, y, z) within a manifold M, the positive (negative)-energy eigenvectors of the 2 x 2 Hamiltonian H = xax +ycy + zoz (where ax,y,z are the Pauli matrices) form a U(l) fiber bundle. For certain M the bundle has non-trivial topology. For example when M = S2 t h e associated bundle has non-zero Chern number indicating that it is topologically non-trivial at the highest level. In this paper we construct a simple 2 x 2 Hamiltonian whose eigen-vector bundle exhibits a more subtle topological non-triviality when .M is a closed three-manifold. This non-trivial topology is characterized by nonzero Chern-Simons invariant.
Twenty years ago I was a graduate student at M.I.T.. Together with many friends in the Boston area, I was often invited to Fred's house for Chinese holidays. I remember vividly that in one occasion Fred entertained us by balancing a women's slipper on his nose. Those parties meant a lot to a young men just arrived in a foreign country. Academically Fred introduced me to the field of statistical mechanics. At that time my thesis work mainly dealt with the electronic structure of semiconductor surfaces. To be frank, I was a little bored with that subject. In a lucky incident I stumbled upon a problem concerning the magnetic properties of spin 1/2 antiferromagnet on a triangular lattice. (Until today this problem is still of considerable interest.) I decided to attack the classical version of this problem, and before long I further simplified the problem by assuming the spins only have two components. It turns out that this problem (classical antiferromagnetic xy model on triangular lattice) is not trivial at all. Fred advised me to do a mean-field calculation first. I took the advice and soon discovered the ground state degeneracy is 2 x oo. Here oo comes from the global spin rotation symmetry and 2 comes from an interesting chirality ordering in the ground state. Together with Fred (and another fellow graduate student and my thesis adviser) we wrote a paper on the mean-field phase diagram of this problem. This work initiated a series of further studies of the critical behavior when the spins become disordered. Recently Fred get interested in the knot theory in statistical mechanics. As to myself, I have been working in the field of quantum statistical mechanics of strongly 55
1908 D.-H. Lee
correlated systems. One day, while teaching the Berry phase to my quantum mechanics class, I encounter the following question. It is well known that the Berry phase of a spin 1/2 in external magnetic field (Eq. (1)) is the integral of the vector potential due to a magnetic monopole. In mathematical language this vector potential is the connection on a non-trivial fiber bundle. The monopole strength is the Chern number. I want to know whether we can define a simple 2 x 2 Hamiltonian whose Berry connection shows zero Chern number but non-zero Chern-Simons invariant. Since Fred's Festschrift takes place in the Nankai Institute which was founded by professor Chern (to whom I have tremendous admiration) I thought this subject is particularly appropriate. Since it's discovery in 1984,x the Berry phase has played an important role in quantum mechanics. For a simple example of the Berry phase, consider the following two by two Hamiltonian H(T) - xax + ycry + zaz = r • a,
(1)
where ox,y,z a r e the three components of the Pauli matrices and x, y, z are real parameters. For a fixed r = (x, y, z) the Hamiltonian in Eq. (1) has two eigenvalues E± = ±y/x2 + y2 + z2 = ±|r|. Let us focus on the eigenvector |^( r ) > associated with the positive eigenvalue for the rest of the paper. The Berry phase induced by an adiabatic evolution of r around a closed loop C is given by 7
= ^da^4(r),
(2)
where 4 ( r ) = \<
^(r)|a^(r) > .
(3)
It turns out that the A^ in Eq. (3) has a geometric interpretation as we shall explain in the following. At a fixed r if one is given a spinor (i.e. a 2-component column vector) satisfying ff(r)Mr)
> = |r||V(r) >
=l,
(4)
one can generate a continuum of other spinors which satisfy Eq. (4) by the transformation h/>(r)>^e*V(r)>.
(5)
This family of \ip{r) > spans an internal space that is invariant under a U(l) group of transformations (Eq. (5)). As r varies through a manifold M. (henceforth referred as the base space) the internal space sweeps out a geometric object called "fiber bundle". Since U(l) leaves the internal space invariant this fiber bundle is a U(l) 56
The Chern-Simons Invariant in the Berry Phase 1909
bundle. In the following we shall refer to such fiber bundle as the eigenbundle of Eq. (1). It turns out that Eq. (3) is precisely the geometric connection on the eigenbundle. 2 The connection A1^ defined above is not unique. Indeed, by performing the transformation |^(r) >—» el$^\ip(r) > we induce a "gauge transformation" on A^: Al-+A\
+ d,>0.
(6)
It is obvious that the Berry phase (Eq. (2)) is gauge invariant. Next we shall focus on two-dimensional base spaces M. that are closed surfaces. It turns out that if Ai encloses the origin (for example M. — S2 = {r; |r| = 1}), it is impossible to choose a gauge in which A1^ is non-singular everywhere. In order to obtain locally non-singular A^ it is necessary to divide M. into a number of (overlapping) patches so that 1) Ab is non-singular in each patch, and 2) in the region where two patches overlap the different A^s differ only by a gauge transformation. Historically this problem was encountered by Dirac when he tried to write down the vector potential in the neighborhood of a magnetic monopole.3 It turns out that under the framework of quantum mechanics, condition 2) requires the strength of the monopole to be quantized. 3 ' 4 In geometry it is known that the non-existence of an everywhere-nonsingular connection is the manifestation of non-trivial topology. In his seminal work S.S. Chern discovered a set of invariants to characterize such non-triviality.5 For the simple case we are considering the invariants reduce to a single number, the Chern number:
e
(7)
-h\M**r*>"
Here Fbv = d^A^ — dvAb is the curvature associated with A* . For the eigenbundle of Eq. (1) it is simple to show that C = 1/2 or 0 depending on whether M encloses the origin. If we interpret FM„ as magnetic field, the above result suggests that C is the total magnetic flux (through M.) produced by a magnetic monopole located at r = 0. In a proof similar but more general than that given in Ref. 3, Chern showed that C should be quantized to values n/2 where n — integer.5 Since C > 1/2 is allowed, it is interesting to ask what kind of Hamiltonian will have C = n/2 (n > 1) eigenbundles. One answer is given by the following (n + 1) x (n + 1) matrix H(T)
(8)
= T-S,
where S = (Sx, Sy, Sz) are the matrices representing the three generators of SU(2) in the spin S = n / 2 representation. For example for n = 2 we have /0
1 ON
/0
57
-i
0\
1910
D.-H. Lee
Sz =
0 0
0
.
(9)
There is another way of modifying Eq. (1) which also leads to C = n / 2 eigenbundle. Interestingly this time we do not need to enlarge the dimension of H. Consider the following 2 x 2 matrix H(r) = /*(r) • a,
(10)
where h(r) is a suitably chosen unit vector field that defines a mapping from M, (a closed two-manifold) to S2. It is known that such mappings can be classified into homotopy classes each labeled by an integer V
[ d2xe^J^.
(11)
JM
Here the Pontryagin form JM„ is given by J^ = -rh • (dah x djk).
(12)
We will later show that by choosing a h(r) with V = n the eigenbundle of Eq. (10) exhibits C = n/2. The Chern number records the highest level of topological non-triviality. When the Chern number vanishes the eigenbundle can still be non-trivial at a more subtle level. Let us consider closed three-manifold in which C = 0 for all closed sub two-manifolds, which implies the absence of monopole. Without monopole the "flux lines" associated with the vector field e,l'/XFu\ form closed loops. Under this condition there is a topological interesting situation in which these flux lines link with one another. It is clear that this class of eigenbundles are topologically distinct from those without linking flux lines. In 1974 Chern and Simons discovered an invariant, the Chern-Simons invariant, that quantifies this more subtle level of topological non-triviality.6 For a closed three-manifold M. the Chern-Simons invariant is given by CS = 1- l fx^J^dvAi.
(13)
4TT JM
We note that when M is closed CS is gauge invariant. The topological information recorded by CS is precisely the linking between the flux lines. The fact that linking is only denned in three dimensions explains why the Chern-Simons invariant requires three dimensional base space. Since there is another level of topological non-triviality, it is natural to ask whether one can modify Eq. (1) so that the eigenbundle exhibits such topology. We shall prove that the Hamiltonian given by Eq. (10) also works so long as /t(r) is chosen appropriately. Now let us restrict ourselves to the case where the base space M. is a simply connected closed three-manifold and r labels the points in it. In such case h(r) is a 58
The Chern-Simons Invariant in the Berry Phase 1911
mapping from a simply-connected closed three-manifold to S2. The work of Hopf shows that such mapping can also be classified into homotopy classes by an integral valued Hopf invariant H. A Hopf map is a smooth configuration of h(r). Due to the fact that there are only two linearly independent directions for d^h, it follows e^%
J„ A = 0.
(14)
As the result the flow lines associated with ^V^JV\ form closed loops. For a nontrivial Hopf map any pair of J-loops link with each other. Because of Eq. (14) there exists a A^. so that • V = -^(0 M i4S-0„Aj;).
(15)
The Hopf invariant is simply the Chern-Simons invariant for A^,7's
i.e.,
U = -i- / dzx^vXAhdvA\. 4?r JM In the rest of the paper we prove the following.
(16)
(i) For closed two - manifold M the eigen bundle of Eq. (10) has C = n/2 if h(r) has V = n. (ii) For closed three — manifold M. the eigen bundle of Eq. (10) has CS = n if h(r) has H = n.
(17)
The proof amounts to show the following identity F%,=4*JIU,.
(18)
The Berry curvature is given by
= i [< d^+\dvip+
>-<
dvil)+\dpil>+ > ] .
(19)
To compute < dliip+\dI,il)+ > — < d„^ + |c^V+ > w e insert a complete set of states i1 = E n = ± IV'n > < V'nl); a
n dtnat
g iv eS
< <9MV+I<W+ > - < dvtl)+\dfj,ii+
>
n=±
=< dpil>+\il>- >< ip-\dvip+ > -[n <-> v\.
(20)
In reaching the last line we have used the fact that < dlitp+\tp+ > < ^)+\dvij}+ > — [/iO v] = 0. To compute < ip-\dvip+ > in Eq. (20) we express the eigenvector of H{T') = H(r + Sr) = H(r) + Sxxd\H in terms of those of H(r) via first order perturbation 59
1912
D.-H.
Lee
theory. Up to the first order in Sx11, we obtain
|*+ > + <^-lf A W + >,^ > Ex. — .EL
|^+ > + <^1^M^> | V ,_
> (21)
Eq. (21) implies that < il>-\dvil>+> =
< ip-\dvh • a\il)+ > (22)
As the result we have
= - [ < tp+ld^h-alip-
> < ip-\duh • a\ip+ >
= ^abc{d^K){dvhb)[<
ip+\ac\tp+ >]
= ^eabc(d^ha)(duhb)hc
= -h • (d^fi x dvh).
-[/JHI/]
(23)
Going from the second to the third line of Eq. (23) we have used the fact that < ip+\d,Ji • < ip+\d„h • S\ip+ > - [ / / -H- v] = 0. Substituting Eq. (20) and Eq. (23) into Eq. (19) we obtain (24)
Ft, = \h • dji x dji = Air J, liV
After establishing Eq. (18) it is simple to prove (i) and (ii) of (17). For (i) the Chern number is given by
-hL**<"*>-\L
dfx
e^J,jlV
V/2.
(25)
Thus V = n implies C = n / 2 . Now let us prove (ii) of (17). Eq. (15) and Eq. (18) imply that \IV
ATTJ,fXV
/iW
(26)
Thus A1^ and Abh differ at most by a pure gauge
A^A^
+ d^. 60
(27)
The Chern-Simons Invariant in the Berry Phase
1913
Since Eq. (13) is gauge invariant when M is a closed manifold we conclude that
cs =
hj
d3xelivXAb
Sx x F
A =tj3
^K ^
= n.
(28)
Thus V. = n implies CS = n. In physics one often encounters the Berry phase when a system posses's both "fast" and "slow" dynamic degrees of freedom. When the fast degrees of freedom are integrated it often produces, as part of the effective action of the slow variables, a Berry phase term that is non-zero even when the slow variables change adiabatically with time. Such term can fundamentally alter the behavior of the slow variables. Here we present an example where the fast degrees of freedom generate an effective action represented by the Hopf invariant of the slow variables. The model is a field theory in 2+1 space-time dimensions. It consists of two fields: 1) a fermion field •0 a n d 2) an unit vector field h(r, t). The Lagrangian density is given as £ = £,/, + £„ - gh • Tpaffapipp A/> =$a{dt
- t*)lpa - -Z— $a(y
£ n =ifi[n] + y|Vn| 2 .
~
iA.ex)2ll>a
(29)
In the above m, g, c, /J, are parameters of the model, Aex is the vector potential of an external magnetic field B, i.e., dxAy — dyAx = B, and 6il/6h = h x dtfi. Physically £,/, describes fermions moving in an external magnetic field, and £ n describes the dynamics of magnetic moments in a ferromagnet. The last term in the first equation is the Zeeman coupling between the fermions and the magnetic moments. By adjusting fi we can tune the density (p) of the fermions so that P = *;r> (30) 00 where 0o = 27r is the Dirac flux quantum and k is an integer. When Eq. (30) is satisfied, the ground state of the fermions is an "integer-quantum Hall liquid" . 9 Let us further assume that g is large so that locally the electron spins align with the direction of ft. Under that condition integrating out the fermion field produces a term J^ J cPxdte^^A^d^A^, which is proportional to the Hopf invariant of the n(r, t). This term has the effect of changing the spins and statistics of solitons (the skyrmions) in the h(r, t) field.7 Acknowledgements: DHL is in debt to Geoffrey Lee for helping him to visualize the dual of the Pontryagin form of the non-trivial Hopf map. DHL is supported by NSF grant DMR 99-71503. References 1. M. Berry, Proc. R. Soc. London, A 392, 45 (1984). 61
1914
D.-H. Lee
2. We note that in order for A^ to be well defined, the reference eigenvector \ip{r) > must be differentiable with respect to r. 3. P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931). 4. C.N. Yang, Ann. of N.Y. Acad, of Sci. 294, 86 (1977). 5. S.S. Chern, Ann. of Math. 47, 85, (1946). 6. S.S. Chern and J. Simons Ann. Math. 99, 48 (1974). 7. F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984). 8. Y-S Wu and A. Zee, Phys. Lett. 1 4 7 B , 325 (1984). 9. For a review see, e.g., "The Quantum Hall Effect", edited by R.E. Prange and S.M. Girvin, (Springer-Verlag, New York 1986.)
62
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1915-1924 © World Scientific Publishing Company
M U T U A L L Y LOCAL FIELDS FROM FORM FACTORS*
ANDREAS FPJNG Institut
fur Theoretische Physik, Freie Universitat Berlin, Amimallee 14, D- 14195 Berlin, Germany E-mail: [email protected]
Received 21 October 2001 We compare two different methods of computing form factors. One is the well established procedure of solving the form factor consistency equations and the other is to represent the field content as well as the particle creation operators in terms of fermionic Fock operators. We compute the corresponding matrix elements for the complex free fermion and t h e Federbush model. The matrix elements only satisfy the form factor consistency equations involving anyonic factors of local commutativity when the corresponding operators are local. We carry out the ultraviolet limit, analyze the momentum space cluster properties and demonstrate how t h e Federbush model can be obtained from the S[/(3)3-homogeneous sine-Gordon model. We propose a new class of Lagrangians which constitute a generalization of the Federbush model in a Lie algebraic fashion. For these models we evaluate the associated scattering matrices from first principles, which can alternatively also be obtained in a certain limit of the homogeneous sine-Gordon models.
1. Introduction One of the most central concepts in relativistic quantum field theory, like Einstein causality and Poincare covariance, are captured in local field equations and commutation relations. In fact this principle is widely considered as so pivotal that it constitutes the base of a whole subject, i.e. local quantum physics (algebraic quantum field theory) 2 which takes the collection of all operators localized in a particular region generating a von Neumann algebra, as its very starting point. On the other hand, in the formulation of a quantum field theory, one may alternatively start from a particle picture and investigate the corresponding scattering theories. In particular for 1+1 dimensional integrable quantum field theories this latter approach has been proved to be impressively successful. As its most powerful tool one exploits here first the bootstrap principle, 3-5 which allows to write down exact, i.e. non-perturbative, scattering matrices. Ignoring subtleties of nonasymptotic states, it is essentially possible to obtain the particle picture from the field formulation by means of the LSZ-reduction formalism.6 However, the question * based on reference 1 63
1916
A. Pring
of how to reconstruct the field content, or at least part of it, from the scattering theory is in general still an outstanding issue. This talk is also devoted to this question in the sense that we provide explicit expressions for operators 0(x) located at x in terms of fermionic Fock fields. Particular emphasis is put on the question whether these operators are really local in the sense that they (anti)-commute for space-like separations with themselves, [O(x),O(y)]=0
(x-y)2<0
for
(1)
and how this property is reflected in the form factor consistency equations. It will turn out that from possible matrix elements the form factor consistency equations select out those which correspond to mutually local operators. We argue that the presence of the factor of local commutativity in these equations is absolutely essential. 2. Determination of form factors Let us assume that there is no backscattering in our model and that we have explicitly determined its two-particle scattering matrix, which can be expressed as a phase in this case. We further presume that the S-matrix results from braiding two particle creation operators ZU6) for stable particles of type fj, with rapidity 9, which obey the Faddeev-Zamolodchikov algebra 7 ZJ(9i)z}(9j)
= Sij^Z^zUOi)
= exp[2niSij(9ij)}Z}(9j)Zl(9i).
(2)
As common we parameterize the two-momentum p by the rapidity variable 8 as p = m(cosh9, sinh#) and abbreviate 8^ :— 9i — Oj. In order to pass from scattering theory to fields, we want to determine the form factors, i.e. the matrix element of a local operator 0(x) located at the origin between a multi-particle in-state and the vacuum F°^-"^(91,...,0n)
= (o(O)Z}li(01),...,Zln(9n))ia.
(3)
We distinguish between the mere matrix element F° and the particular ones which also solve the consistency equations in 2.1, in which case we denote them as F®. 2.1. Form factors from consistency
equations
Various schemes have been suggested to compute the objects in equation (3). One of them consists of solving a system of consistency equations which have to hold for the n-particle form factors based on some natural physical assumptions, like unitarity, crossing and bootstrap fusing properties 8 " 11 Fn F^-^(9l F^-^(91
X1
(...,9i,6j,...)
= Fn
J
" (...,6j,8i,...)SnitJ,.{9ij),
+ 2m,...,9n)=^iF^-^(92,...,6n,91) + A,... ,9n + A) = esXF°^-^(9u 64
(4) ,
...,9n) ,
(5) (6)
Mutually Local Fields from Form Factors
Res F°l¥^~»»(9
+ in,90,9v
1917
. . » „ ) = * ( l - 7 ? ft W * < « ) ) * f ^"''"(fli- • .0„).(7)
e
-*e°
1=1
Here s is the Lorentz spin of the operator 0 and A is an arbitrary real number. We omitted here the so-called bound state residue equation, which relates an (n + 1)to an n-particle form factor, since it will be of no importance to the explicit models we consider. We stress the importance of the constant 7 ^ , the factor of so-called local commutativity defined through the equal time exchange relation of the local operator 0(x) and the field O^y) associated to particle creation operators Z^ Ofi(x)0(y)=^0(y)Ofi(x) 0
for
x1 > y1,
(8)
1
with x^ = (a; , a; ). This factor carries properties of the operator and not just of the Z^'s. An immediate consequence of its presence is that a frequently made statement has to be revised, namely, that (4)-(7) constitute operator independent equations, which require as the only input the two-particle scattering matrix. Here we demonstrate that apart from ± 1 , which already occur in the literature, this factor can be a non-trivial phase. Thus the form factor consistency equations contain also explicitly non-trivial properties of the operators. To solve these equations at least for the lowest n-particle form factors is a fairly well established procedure, but it still remains a challenge to find closed analytic solutions for all n-particle form factors. 2.2. Direct computation
of matrix
elements
The most direct way to compute the matrix elements in (3) is to find explicit representations for the operators Z^O) and 0(x). To represent the former operator is known in complete generality for theories not involving backscattering. A representation for these operators in the bosonic Fock space was first provided in Ref. 12 d0 , <J iI (0-0')aJ(0 , )a,(0 / ) al(0). J where the a's satisfy the usual fermionic anti-commutation relations
z\{e)
exp
{ai(e),aj(e')}=Q
and
{ai(0),aj(0')} = 2*5^5(0 - 9').
(9)
(10)
Having obtained a fairly simple realization for the Z-operators, we may now seek to represent the operator content of the theory in the same Fock space. Hitherto, it is not known how to do this in general and we have to resort to a study of explicit models at this stage. 3. Complex free Fermions Let us consider N complex (Dirac) free Fermions described as usual by the Lagrangian density £
FF = E l l ^
^ 65
" m°W°
•
(11)
1918
A. Fring
We define a prototype auxiliary field dOdO'
+na(6,0'
- »7r) ( 4 ( 0 ) a a ( 0 ' K ( p - p , ) - x - O a ^ o t ^ J e - * ^ ) ' * ) ]
(12)
and intend to compute the matrix element of general operators composed out of these fields 0**{x) = :e*-(*>:, Ox«{x) = : f •^(aa{p)e'ip"-x J
+ 4(p)e*-- x )e>£< x >:.
2n
(13)
Pa
Employing Wick's first theorem, we compute 1 F?f]nX&a(0i,...,e2n)
= [dei--\d62nflKa(0'2i-MdetV^ J
2n+l
n
-
,
(14)
i=i
'(*i, • • • ,<W) = / ^ • "nf2n+1 n^(^,^ + i)detl> 2 " +1 ,(15)
where T^ is a rank ^ matrix whose entries are given by ^.-cos2[(i-i)7r/2]^;-^),
l
(16)
Note that 0XK (x) and (f)x" (x) are in general non-local operators in the sense of (1). At the same time F® is just the matrix element as defined on the r.h.s. of (3) and not yet a form factor of a local field, in the sense that it satisfies the consistency equations (4)-(7), which imply locality of O. A rigorous proof of this latter implication to hold in generality is still an open issue. Let us now specify the function K. The free fermionic theory possesses some very distinct fields, namely the disorder and order fields fia(x) = :eu-W:
and
aa(x) = $a(x)pa{x)-.,
a = 1,2,
(17)
respectively. We introduced here the fields
We compute 1 the integrals in (14) and (15) for this case and obtained a closed expression for the n-particle form factors of the disorder and order operators
F2^nxl\e1,...,
e2n) = ( - i ) » F £ | n x 2 2 ( - 0 i , • • •. - 0 * 0
* £ | n x I 1 H > i , • • •, -fcn) = (-i) n i^ n s|nx22 (fli, • • •, M = in2n-1an{x1, p(Ti|l(nxll)//,
o
\ — (
x3, • . . , x2n-i)Bn,n,
i'\n P
a
r,<7i|l(nxll)/ n n \ _ ( -, \n W7 2 |2(rax22), r, ^2n+l ( - t / l , - - - , - W 2 n + l j - ( - 1 ) -T 2 n+1 \?l, • • •
Q
v
„ N ,V2n+\)
= in2n-1
(19)
(20)
Mutually Local Fields from Form Factors
1919
with l l l < i < j < n ( ^ 2 i - l ~ % ) - l ) lIl
U
j)
Associated with the particles and anti-particles we introduced here the quantities Xi = exp(#,) and Xi = exp(0j), respectively. The variable Ui can be either of them. We also employed the elementary symmetric polynomials ak(x\,..., xn). The remaining form factors are zero due to the U(l)-symmetry of the Lagrangian. One may easily verify that the expressions (19) and (20) indeed satisfy the consistency equations (4)-(7) with */£* = — 1. and j ^ a = 1 for a = 1,2. We also compute1 the form factors associated to the trace of the energy-momentum tensor Fif""| a' o"(M) P , 0) = F, , 9 ) = -2-Kiml *f""| 'a"a>(M) -2*i» a sinh -«»• ^
—,
(22)
which plays a distinct role in the ultraviolet limit. 4. The Federbush M o d e l The Federbush model 13 was proposed forty years ago as a prototype for an exactly solvable quantum field theory which obeys the Wightman axioms. 14 It contains two different massive particles Wi and \I>2- A special feature of this model is that the related vector currents Jg = ^fa^^^a, a e {1,2}, whose analogues occur squared in the massive Thirring model, enter the Lagrangian density of the Federbush model in a parity breaking manner
- J2a=1
£F
2
*«(*7M«M
- ™<0*a " 27rAeM„ J{* J J
(23)
due to the presence of the Levi-Civita pseudotensor e. The scattering matrix was found to be 14 - 16 / -
1
1
1
1
e2-7TiA
2niX
\e~
e-2TuA
2niX
e
e
-2«A
e27TiA
p27T«A
_ — 27HA
j
j
1
1
\ '
'
'
/
For the rows and columns we adopt here the ordering {1,1,2,2}. In close relation to the free fermionic theory one may also introduce the analogue fields to the disorder and order fields in the Federbush model **(*) = :exp[fi*(x)]: = \ exV{-2^i\a(x)}\
E
" w = : / ^k{aa{p)e~iPa'x
+
(25)
^(p)^--*) *iw : -
(26)
where the /t-function related to Q is MO a<\ s.2t * nn _ »sin(7rA)e (0,9)-n (-6,-9)2coshl(e_9f)
K
67
^B'e^
•
(27)
1920
A. Fring
The last equality in (25) was found by Lehmann and Stehr, 15 who showed the remarkable fact that the operator $„(a;) can be viewed in two equivalent ways. On one hand it can be denned through triple ordered free Bosons <j>a{x), defined as •eK0- _ e «0J ^eK0^ for K D e i n g some constant, and on the other hand by means of a conventional fermionic Wick ordered expression. We compute 1 the following equal time exchange relations for a, ft = 1,2 1>a(x)H(v) = *p(v)4>a(x) e^-VM-W1-*1),
(28)
x
-rl>a(x)Z%{y) = V p(y)fl>a(x) e*"H-i)>xs.,e&-S);
(29)
**(x)ty(y) = *fo)**(x)
(30)
E*(x)E^(y) = S^(y)E*(x)
2 e
^ ~ ^ ^
.
(31)
where @(x) is the Heaviside step function. With the relevant exchange relations at our disposal, we can, according to (8), read off the factors of local commutativity for the operators under consideration = - 7 ? = e**iW*5<">
J}
and
7*'
= - 7 ? = e-2*iWM'">
.
(32)
Proceeding again in the same way as in the previous section, we obtain as closed expressions for the n-particle form factors F
2n \,Xi,X2...X2n-l,X2n) A _ r , * f | n x l l ,_
F2^
= {-l)F27? (Xt, X2 . . . X2n-1, A . , .„ _* |nx22/_ _
= (-1) F 2n 2
(xi,x2...x2n-1,x2n) n
n
1
X2n) = ,.
{xi,x2...x2n-i,x2n)
=
B
i 2 " 8in (7rA)(Tn(«i . . .X2n-l) A + *<7„(*2 • • • X2n)>~XBn,n, pV$\l(nxll)m
„
, _ ,
^„pS-A|2(nx22),fl
*fc+1
i n
x+
(2i) an(x2...x2n) 2 <7 n (a;i...x 2 „ + i) A 2
T-T , "
^ y^ V ^ )
2
,_
ft
^£2^*11(71x11)^ EjA|l(nxll),fl . _ , ^ n,„£ 7,sJ|2(nx32)^ 3S A |2(nx22),, „ ,__^ a C/ *2n+l l l ) - - - ^ 2 n + l j - l - l j -f*2n+l (01,---,P2
,
sin"(7rA) 2 {Xj-Xl)
II j
We may now convince ourselves, that the expressions for i^n
(33)
,
.
"" indeed satisfy
om the consistency equations (4)-(7). However, the expressions of i^n+i * y sat" isfy the consistency equations (4)-(7) for A = 1/2. This reflects the very important fact that E^(cc) is only a mutually local operator for this value of A, see equation (31), unlike $a(x) which is mutually local for all value of A. Thus, the equations (4)-(7) select out solutions corresponding to operators which are mutually local. The form factors related to the trace of the energy-momentum tensor turn out to be the same as the ones for the complex free Fermion.
Mutually Local Fields from Form Factors
1921
5. Momentum space cluster properties An interesting operator related property which the form factors satisfy is the momentum space cluster decomposition lim i f + ( ( 0 i . . . 6k, dk+1 + A . . . 6k+l + A) = Fffa
... 6k)F?"(0k+1...
6k+l) ,(35)
A—>oo
Writing instead of the matrix elements only the operators, we obtained 1 formally the following decomposition <6 A
> s>* x *a
a
a
><
Ma
>S
(36)
together with the equations for a ^ a. This means the stated operator content closes consistently under the action of the cluster decomposition operators. We also observe that non-selfclustering, i.e. O ^ O' ^ O", is possible. Unlike the self-clustering, which can be explained for the bosonic case with the help of Weinberg's power counting argument, this property is not yet understood from general principles. 6. Lie algebraically coupled Federbush models The Federbush model as investigated in the previous section only contains two types of particles. In this section we propose a new Lagrangian, which admits a much larger particle content. The theories are not yet as complex as the homogeneous sine-Gordon (HSG) models, but they can also be obtained from them in a certain limit such that they will always constitute a benchmark for these class of theories. Let us consider I x £-real (Majorana) free Fermions ipa^{x), now labeled by two quantum numbers 1 < a < £, 1 < j < I and described by the Dirac Lagrangian density £FF- We perturb this system with a bilinear term in the vector currents
£
I M
£CF = E E *«,i(H 3M " ™aj)**J - ^ a=l j=l
£
£
E
E
<jJ^K$
>
(37)
a,b=lj,k=l
and denote the new fields in £ C F by ^a,j- Furthermore, we introduced £2 x t2 dimensional coupling constant dependent matrix h?ab, whose further properties we leave unspecified at this stage. We computed1 the related S-matrix to
S£ = - e * A £ .
(38)
where due to the crossing and unitarity relations we have the constraints A £ = -A*j + 2Z
and
A% = A g + 2Z
(39)
on the constants A. Taking A3ab = 2Xab£jkIjkK^, with K, I being the Cartan and incidence matrix, respectively, provides the limit of the HSG-models. 69
1922
A. Pring
7. The ultraviolet limit The ultraviolet Virasoro central charge of the theory itself can be computed from the knowledge of the form factors of the trace of the energy-momentum tensor 17 by means of the expansion oo
n
i^
oc
2
dOi...dOn
rn
[Oi,...,On)
(40)
W J L
J
(ELi *»* cosh^^
In a similar way one may compute the scaling dimension of the operator O from the knowledge of its n-particle form factors 18 oo
sp
1 v~^ v ^
=
uv
°°
°°
f
f
2
" <°> h.h.L''
L
xFr>1""'i-(fli,...,«„)
d6\...ddn n[ n
(SLI™*-sh^)2
^
(F„
O | M
"(0I,...A))*
.
(41)
In general the expressions (40) and (41) yield the difference between the corresponding infrared and ultraviolet values, but we assumed here already that the theory is purely massive such that the infrared contribution vanishes. Evaluating these formulae, we obtain cuv = 2
and
A"«
uv
(42)
16
for the complex free Fermion and cuv — 2
and
A " £l/2
A*"
A
"
(43)
— A a —
-
ZA U V -
£1/2
4
.
for the Federbush model Note, that AUv = AUv = 1/16, which is the limit to the complex free Fermion. Yet more support for the relation between the 5?7(3)3-HSG model and the Federbush model comes from the analysis of A = 2/3, for which the 5C/(3)3-HSG S-matrix is related to the one of the Federbush model. In that case we £2/3
£2/3
obtain from (43) the values AUv = AUv = 1/9, which is a conformal dimension occurring in the 577(3)3-HSG model. Thus precisely at the value of the coupling constant of the Federbush model at which the S'J7(3)3-HSG S-matrix reduces to the 5 F B , the operator content of the two models overlaps. 8. Conclusions We summarize our main results: We computed explicitly closed formulae for the n-particle form factors of the complex free Fermion and the Federbush model related to various operators. We carried out this computations in two alternative ways: On the one hand, we represent explicitly the field content (12) as well as the particle creation operators (9) in terms of fermionic Fock operators (10) and computed thereafter directly the 70
Mutually Local Fields from Form Factors 1923 corresponding m a t r i x elements. On the other h a n d we verified t h a t these expressions satisfy t h e form factor consistency equations only when the operators under consideration are mutually local, i.e. satisfying (1). It is crucial that the consistency equations contain the factor of local commutativity 7 ^ as defined in (8). Our analysis strongly suggest that the form factor consistency equations select out operators, which are mutually local in the sense of (1). O u r solutions turned out to decompose consistently under the m o m e n t u m space cluster property. This computations constitute next t o the ones in Refs. 19,20 the first concrete examples of non-selfclustering, i.e. O —> O' x O" in the sense of (36). F u r t h e r s u p p o r t for the identification of the solutions of (4)-(7) with a specific operator was given by an analysis of the ultraviolet limit. We d e m o n s t r a t e d how the scattering matrix of t h e Federbush model can be obtained a s a limit of the 577(3)3-HSG scattering m a t r i x . This "correspondence" also holds for t h e central charge, which equals 2 in b o t h cases, and the scaling dimension of t h e disorder operator at a certain value of the coupling constant. We p r o p o s e d a Lie algebraic generalization of t h e Federbush models, by suggesting a new t y p e of Lagrangian. We evaluate from first principles the related scattering matrices, which can also be obtained in a certain limit from the HSG-models. We expect t h a t the construction of form factors by means of free fermionic Fock fields c a n b e extended to other models by characterizing further t h e function K.
References 1. O.A. Castro-Alvaredo and A. Fring, hep-th/0107015, to be published in Nuclear Physics B 2. R. Haag, Local Quantum Physics: Fields, Particles, Algebras 2-nd revised edition (Springer, Berlin, 1996). 3. B. Schroer, T.T. Truong and P. Weisz, Phys. Lett. B 6 3 , 422 (1976). 4. M. Karowski, H.J. Thun, T.T. Truong and P. Weisz, Phys. Lett. B67, 321 (1977). 5. A.B. Zamolodchikov, JETP Lett. 25, 468 (1977). 6. H. Lehmann, K. Symanzik and W. Zimmermann, NUOVQ Cimento 1, 205 (1955). 7. A.B. Zamolodchikov and ALB. Zamolodchikov, Ann. of Phys. 120 (1979) 253; L.D. Faddeev, Sov. Set. Rev. Math. Phys. C I , 107 (1980). 8. M. Karowski and P. Weisz, Nucl. Phys. B139, 455 (1978). 9. F.A. Smirnov, Form factors in Completely Integrable Models of Quantum Field Theory, Adv. Series in Math. Phys. 14 (World Scientific, Singapore, 1992). 10. V.P. Yurov and Al.B. Zamolodchikov, Int. J. Mod. Phys. A6, 3419 (1991). 11. H. Babujian, A. Fring, M. Karowski and A. Zapletal, Nucl. Phys. B538 [FS], 535 (1999). 12. A. Fring, Int. Jour, of Mod. Phys. 11, 1337 (1996). 13. P. Federbush, Phys. Rev. 121, 1247 (1961). 14. A.S. Wightman, High Energy Interactions and Field Theory, Cargese Lectures, ed. M. Levy, (Gordon and Breach, New York, 1966). 15. H. Lehmann and K. Stehr, The Bose field structure associated with a free massive Dime field in one space dimension, DESY Preprint 76/29, (1976). 16. B. Schroer, T.T. Truong and P. Weisz, Ann. of Phys. 102, 156 (1976). 17. A.B. Zamolodchikov, JETP Lett. 43, 730 (1986). 71
1924 A. Fring 18. G. Delfino, P. Simonetti and J.L. Cardy, Phys. Lett. B387, 327 (1996). 19. O.A. Castro-Alvaredo, A. Fring, C. Korff, Phys. Lett. B484, 167 (2000). 20. O.A. Castro-Alvaredo and A. Fring, Nucl. Phys. B 6 0 4 , 367 (2001); Phys. Rev. D63, 21701 (2001); Phys. Rev. D64, 85007 (2001).
72
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1925-1930 © World Scientific Publishing Company
D E F O R M A T I O N QUANTIZATION: IS C i N E C E S S A R I L Y SKEW?
CHRISTIAN FRONSDAL Physics Department,
University of California,
Los Angeles GA 90095-154% USA
Received 5 November 2001 Deformation quantization (of a commutative algebra) is based on the introduction of a new associative product, expressed as a formal series, / * g = fg + ^ hnCn(f,g). In the case of t h e algebra of functions on a symplectic space the first term in the perturbation is often identified with the antisymmetric Poisson bracket. There is a widespread belief t h a t every associative *-product is equivalent to one for which Gi(f,g) is antisymmetric and that, in particular, every abelian deformation is trivial. This paper shows t h a t this is far from being t h e case and illustrates the existence of abelian deformations by physical examples.
1. Introduction First, a very brief review of quantization: • H. Weyl looked at quantization as a one-to-one correspondence between functions on phase space on the one hand, and operators in a Hilbert space on the other: 1 function on phasespace = / H* W(f) = operator.
(1)
• Taking over Weyl's correspondence, Moyal2 proposed an autonomous formulation of quantum mechanics, in terms of functions on phase space endowed with a new, non-commutative product, called a *-product, namely f*g
= W-\W{f)W{g)).
(2)
• Moyal's idea was incorporated into the new, deformation theory approach to quantization, 3 where one studies more general associative *-products viewed as formal series in a parameter
/*<7 = X> n C»(/,ff),
C0(f,g) = fg,
(3)
ra=0
and usually Ci(f,g) 73
= \{f,g}.
(4)
1926
C. Fronsdal
The bracket {,} is the Poisson bracket. Such deformations "in the direction of the Poisson bracket" play a basic role in Drinfel'd's approach to quantum groups. 4 The existence and classification of *-products on an arbitrary symplectic manifold were placed in a nice geometric context by the approach of Fedosov;5 see also Ref. 6. • Recently, M. Kontsevich has shown 7 that *-products exist on arbitrary Poisson manifolds. See also Ref. 8. This is a significant achievement that is currently exciting much interest among mathematicians. This summary suggests that the first order term, Ci(f,g), is always antisymmetric in its two arguments. But there is a footnote to the story, namely: • Geometric, Souriau-Kostant quantization on co-adjoint orbits, 9 ' 10 and the generalization of this method within the *-product-deformation approach (see for example Ref. 11) is a parallel development. As we shall see, the most interesting case, that of quantization on a singular orbit, is characterized by a *-product for which C\ (/, g) is not skew. We also cite the program of quantization of Nambu mechanics that ran aground on the belief that every abelian *-product is trivial. 12 This is mistaken, and for two independent reasons; first, because it is false on varieties with singularities and second, because it holds only to first order in the deformation parameter. 2. Quantization on algebraic varieties On the manifold R, with global coordinates JCI, • • •, xn, consider the algebra A=Q[xi,---,xn]
(5)
of polynomials in n variables. On this algebra, consider an associative deformation of the ordinary product (the commutative product of functions) in the form oo
/*<7=5> n C„(/, 5 ),
C0(f,g)=fg.
(6)
n=0
Associativity, to first order in h, is the statement that dC\ = 0, where d is the Hochschild differential, dC(f, g, h) = fC(g, h) - C(fg, h) + C(f, gh) - C(f, g)h.
(7)
Triviality of the *-product (see below) is also expressed in terms of the Hochschild differential: The above *-product is trivial to first order if there is a one-cochain E\ such that C\ = dE\, where dE(f, g) := JE{g) - E(fg) + E(f)g.
(8)
The following theorem tells us that, to first order in h, all abelian *-products on IR™, and indeed all abelian *-products on any smooth manifold, are trivial. 74
Deformation Quantization: Is C\ Necessarily Skew? 1927
Theorem: (Hochschild, Kostant, Rosenberg13) If A is the generated algebra of functions on a smooth manifold then the space Hn(A) is the space of alternating n- forms. This implies that, to first order in h, every *-product on a smooth manifold is equivalent to one for which C\ is antisymmetric.
3. Escape route number one Let us consider algebraic varieties with singularities, for example the following. M = U/(x2-y2), Expand / G A, formula
f(x,y)
A=(£[x,y]/(x2-y2).
(9)
= f\(x) + yf2{x) and define a deformed product by the
f*g
= fg + hf2g2-
(10)
It is associative to all orders, and not trivial. Proof of non-triviality: Ah=(E[x,y)/(x2-y2
+ fi).
(11)
If h is real and not zero, then this is the coordinate algebra on a smooth manifold, clearly not isomorphic to the coordinate algebra of the cone x2 = y2. A large class of examples is provided by algebraic varieties, M = JRn/R,
R = a set of polynomial relations, A=(E[xi,---,xn]/R.
(12) (13)
The calculation of cohomology is often fairly simple and rests on the fact that the Hochschild cohomology of A is equivalent to its restriction to linear, closed chains. (See Ref. 14.) In our example, the closed, linear chains are x Ay,
and
d(x A y) = xy — yx = 0,
x®x — y®y,
(14)
d(x (g> x — y <8> y) = x2 — y2 = 0.
(15)
The 2-cochain C is closed if it is symmetric and it is exact if C{x ®x — y®y) G A. A representative of the unique non-trivial equivalence class of 2-forms is C(xAy)=Q,
C(x®x-y®y)
= kE®-
{0}.
(16)
Besides Nambu mechanics, this has applications to the problem of quantization on coadjoint orbits. 75
1928
C. Fronsdal
Quantization
on coadjoint
orbits, an example
The algebra is the universal enveloping algebra of sl(2,R), basis z ^ a ^ ^ o - The unique singular orbit is Q := x\ + x\ — XQ = 0. Invariant quantization is obtained via the Weyl-type correspondence: Xj * X{ = = iltijfcXk)
X% * Xj
P*(a)=Pn(a),
a= £
a'xi,
V '/
(18)
i=l,2,0
where Pn are Legendre polynomials and P* are the same symmetric ^-polynomials, and the assignment of a value to the Casimir operator Q* : X\ * X\ + X2 * X2 - XQ * XQ I-> Tll{l + 1).
(19)
For n = 2, on the singular orbit Q = 0, -{xi * XJ + i,j) - -SijQ* = XiXj,
(20)
Xi * Xj = x^j + h i -ztijkXk + -^Sijl(l + 1) J.
(21)
and thus
This makes use of the cohomologically nontrivial symmmetric 2-cochain C?,(xi, Xj] oc Sij as well as the more familiar antisymmetric cochain. 4. Escape route number two In the examples given, associativity is satisfied to all orders with Cn = 0 , n > 1. Non-triviality is verified to lowest order and of course cannot be changed in higher orders. In the example that follows we shall again have associativity to all orders, and triviality in the lowest order, but this does not imply triviality in higher orders. Triviality of a *-product is the statement that there is a map E : A —¥ A, of the form oo
E(f) = f + Y,nnEn(f),
(22)
n=l
such that f*g
= E-1(E(f)E(g)).
(23)
To first order in Ti this says that C1(f,g)
= E1(f)g-E1(fg)+fE1(g)=:dE1(f,g).
(24)
Here is an example of an abelian *-product that is trivial to first order but nontrivial when taken to all orders. 76
Deformation Quantization: Is C\ Necessarily Skew? 1929 Let M = Minkowsky space, A = (E[xi, • • •, £4]. Decompose / € A into even and odd parts.: / = ( / + , / _ ) , a n d define: = fg- px2f-9-
f*9
= (f+9+
+ f-9-0-
~ Px2),
f+9- + /-
(25)
Now, let M' = 3 + 2 - d i m e n s i o n a l anti-De Sitter space, more precisely the cone in R 5 : M' = H5/(px2+y2-l),
(26)
and A' =(E[x1,---,x4,y}e/(px2+y2
-I),
(27)
e
where [...] means polynomials of even order. Decompose / e A' as follows, / = f+(x)+yf-(x), then fg = f+g+
+ / _ 5 _ ( 1 - px2) + y(f+g.
+ /_
(28)
Therefore, t h e deformed, ^-product algebra of functions o n Minkowski space is isomorphic to the o r d i n a r y algebra of even functions on AdS. But A and A' are not isomorphic and the *-product is therefore not trivial. (For more details see Ref. 14.) It seems possible t h a t this provides a new approach t o physics in AdS that could help to overcome some of t h e problems of interpretation. References 1. H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, New York, 1931. 2. J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc. 45 (1949) 99-124. 3. F. Bayen, C. Fronsdal, M. Flato, A. Lichnerowicz and D. Sternheimer, Quantum Mechanics as a Deformation of Classical Mechanics, Annals of Physics 111 (1978) 61-110 and 111-151. 4. V.G. Drinfel'd, Quantum Groups, Proc. International Congress of Mathematicians 1986 (American Mathematical Society, 1987), pp. 798-820. 5. B.V. Fedosov, A simple geometrical construction of deformation quantization, J. Diff. Geom. 40 (1994) 213-238, and earlier papers. 6. H. Omori, Y. Maeda and A. Yoshioka, Existence of closed star products, Lett. Math. Phys. 26 (1992) 285-294. 7. M. Kontsevich, Deformation quantization of Poisson manifolds, q-alg/9709040; Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999) 35-72. 8. D.E. Tamarkin, Another proof of M. Kontsevich' formality theorem for Rn, math.QA/9803025. 9. J.M. Souriau, Structures des Systemes Dynamiques, Dunod, Paris, 1970. 10. B. Kostant, Quantization and unitary representations, Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics 170, 87-208, Springer-Verlag, Berlin, 1970. 11. C. Fronsdal, Some ideas about quantization, Rep. Math. Phys. 15 (1978) 113. 12. G. Dito, M. Flato, S. Sternheimer and L. Takhtajan, Deformation Quantization and Nambu Mechanics, Comm. Math. Phys. 183 (1997) 1-22. 13. G. Hochschild, B. Kostant and A. Rosenberg, Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962) 383-408. 77
1930
C. Pronsdal
14. C. Pronsdal, Harrison Cohomology and Abelian Algebraic Varieties, hep-th/0109001
78
Deformation
Quantization
on
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1931-1941 © World Scientific Publishing Company
GENERALIZED GINSPARG-WILSON A L G E B R A A N D I N D E X THEOREM ON THE LATTICE
KAZUO FUJIKAWA Department
of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan E-mail: [email protected]. ac.jp
Received 19 November 2001 Recent studies of the topological properties of a general class of lattice Dirac operators are reported. This is based on a specific algebraic realization of t h e Ginsparg-Wilson relation in the form 75(75.0) + (750)75 = 2a 2 f c + 1 (7s.D) 2 f c + 2 where k stands for a nonnegative integer. The choice k = 0 corresponds to the commonly discussed GinspargWilson relation and thus to the overlap operator. It is shown t h a t local chiral anomaly and the instanton-related index of all these operators are identical. T h e locality of all these Dirac operators for vanishing gauge fields is proved on the basis of explicit construction, but the locality with dynamical gauge fields has not been fully established yet.
1. Introduction Recent developments in the treatment of fermions in lattice gauge theory are based on a hermitian lattice Dirac operator 75.D which satisfies the Ginsparg-Wilson relation1 j5D + D75 = 2aDlhD
(1)
where the lattice spacing a is utilized to make a dimensional consideration transparent, and 75 is a hermitian chiral Dirac matrix. An explicit example of the operator satisfying (1.1) and free of species doubling has been given by Neuberger. 2 The relation (1.1) led to an interesting analysis of the notion of index in lattice gauge theory.3 This index theorem in turn led to a new form of chiral symmetry, and the chiral anomaly is obtained as a non-trivial Jacobian factor under this modified chiral transformation.4 This chiral Jacobian is regarded as a lattice generalization of the continuum path integral. 5 The very detailed analyses of the lattice chiral Jacobian have been performed.6 It is also possible to formulate the lattice index theorem in a manner analogous to the continuum index theorem. 7 " 9 An interesting chirality sum rule, which relates the number of zero modes to that of the heaviest states, has also been noticed. 10 See Refs. 11 for reviews of these developments. We have recently discussed the possible generalization of (1.1) and its implications.12 To be specific, we have discussed a generalization of the algebra (1.1) in 79
1932 K. Fujikawa
the form 75(75^) + (75^)75 = 2a 2fe+1 ( 75 L>) 2fc+2
(2)
where k stands for a non-negative integer and k = 0 corresponds to the ordinary Ginsparg-Wilson relation. When one defines H = 75a£>
(3)
75# + # 7 5 = 2#2fc+2
(4)
T5H + HT5 = 0
(5)
(1.2) is rewritten as
or equivalently
where we defined
rs = y6-H2k+1.
(6)
Note that both of H and Ts are hermitian operators. It has been shown that all the good topological properties of the overlap operator 2 is retained in this generalization. 12 ' 13 The practical applications of this generalization are not known at this moment. We however mention the characteristic properties of this generalization: The spectrum near the continuum configuration is closer to that of continuum theory and the chiral symmetry breaking terms become more irrelevant in the continuum limit for k > 1. The operator however spreads over more lattice points for large k. 2. Representation of the general algebra We first discuss a general representation of the algebraic relation (1.5). The relation (1.5) suggests that if Hcj)n = a\n4>n,
(„,„) = 1
(7)
with a real eigenvalue a\n for the hermitian operator H, then H(T5(j)n) = - a A n ( r 5 ^ „ ) .
(8)
Namely, the eigenvalues A„ and — A„ are always paired if A„ ^ 0 and (T$(/>„,, T5(j)n) ^ 0. We also note the relation, which is derived by sandwiching the relation (1.4) by (<£n,750n) = (aA n ) 2 f e + 1
for
A„^0.
(9)
Consequently |(aA„) 2fe+1 | = | ( ^ , 7 5 ^ ) l < !I<MI|75<M - I-
(10)
Namely, all the possible eigenvalues are bounded by |A„I < \80
(11)
Generalized Ginsparg-Wilson Algebra and Index Theorem 1933
We thus evaluate the norm of Y^n (T5<j>n, T5<j>n) = (n, (75 - # 2ft+1 )(75 - H2k+1)n) = (4>n, (1 -
ff^+S
-
75 ff
2fe+1
+
H^k+V)4>n)
= [l-(aA„) 2 ( 2 f c + 1 ) ] = [1 - (aA„)2][l + (aA n ) 2 + ... + (aA„)4fc]
(12)
where we used (2.3). By remembering that all the eigenvalues are real, we find that n is a "highest" state T50n = 0
(13)
[l-(aA„)2] = (l-aA„)(l + a A n ) = 0
(14)
only if
for the Euclidean positive definite inner product ((f>n,4>n) = E i ^ l W ^ W ' We thus conclude that the states <j>n with A„ = ± ^ are not paired by the operation T^4>n and 75-D<^„ = ±-„,
75<£n = ±<j>n
(15)
respectively. These eigenvalues are in fact the maximum or minimum of the possible eigenvalues of H/a due to (2.5). As for the vanishing eigenvalues H<j>n = 0, we find from (1.4) that # 7 5 0 n = 0, namely, H[(l ± 75)/2]„ = 0. We can thus choose 7 5 D ^ n = 0,
75n = n = -„.
(16)
To summarize the analyses so far, all the normalizable eigenstates 4>n of j5D = H/a are categorized into the following 3 classes: (i) n± ("zero modes"),
75-D„ = 0, 75^>n = ±n,
(17)
(ii) N± ("highest states"),
75_D„ = ±-„,
7 5 0„ = ±
(18)
(hi)"paired states" with 0 < |An| < 1/a, •y5Dn = \n(j>n,
-y5D{r5(l)n) = -Xn(rb(j)n).
Note that T5(T5n) a n for 0 < |A„| < 1/a.
81
(19)
1934
K.
Fujikawa
We thus obtain the index relation 3 ' 4 Trr5 = 5Z(^n,r50n) 71
= ^
(<£n,r 5 n )+
A„=0
^2
(<^n,r50n)+
0<|A„|
^2
(0n>r50n)
|A„|=l/a
A„=0
= £
(0n,(75 - # 2 * + 1 ) 0 n )
A„=0
A„=0
= n+ — n_ = index
(20)
where n± stand for the number of normalizable zero modes with 7s(/>n = ±^>n in the classification (i) above. We here used the fact that T$(j)n = 0 for the "highest states" and that (j>n and T5n are orthogonal to each other for 0 < |A n | < 1/a since they have eigenvalues with opposite signatures. On the other hand, the relation Tr75 = 0, which is expected to be valid in (finite) lattice theory, leads to ( by using (2.3)) TT75 = ^ ( < / > n , 7 5 < £ n ) n
= 5 3 (0">750m) + 5 3 (^"'Ts^n) A„=0
A„#0 fl
= n+ - n_ + J2 ( A n ) 2fe+1 = 0.
(21)
A„#0
In the last line of this relation, all the states except for the "highest states" with A„ = ± l / o cancel pairwise for An ^ 0. We thus obtain a chirality sum rule 10 n+ + N+ = n_ + N_
(22)
where N± stand for the number of "highest states" with 75n = ±^>n in the classification (ii) above. These relations show that the chirality asymmetry at vanishing eigenvalues is balanced by the chirality asymmetry at the largest eigenvalues with I Ara | = 1/a- It was argued in Ref. 4 that N± states are the topological (instantonrelated) excitations of the would-be species doublers. We have thus established that the representation of all the algebraic relations (1.2) has a similar structure. In the next section, we show that the index n+ — n_ is identical to all these algebraic relations if the operator 75D satisfies suitable conditions. 82
Generalized
Ginsparg-Wilson
Algebra and Index Theorem
1935
3. Chiral Jacobian and the index relation The Euclidean path integral for a fermion is defined by
f VxjiViP exp[ J %j>Dil)\
(23)
where $Lhl> = £ $(x)D(x, / •
y)^{y)
(24)
x,y
and the summation runs over all the points on the lattice. The relation (1.5) is re-written as 75 r 5 75£>
+ DT5 = 0
(25)
and thus the Euclidean action is invariant under the global "chiral" transformation4 'tp(x) -^{jj'(x) = •Jpjx) + i y ^ •ip(z)ej5r5(z,
x)^5
Z
1>(v) ->•tf(y)= V»(v) + * J^ e T 5 (w»w)1>W
(26)
w
with an infinitesimal constant parameter e. Under this transformation, one obtains a Jacobian factor Vtjj'Vtp' = JV^Vip
(27)
with J = exp[-2iTrer 5 ] = exp[-2ie(n+ -
ra_)]
(28)
where we used the index relation (2.14). We now relate this index appearing in the Jacobian to the Pontryagin index of the gauge field in a smooth continuum limit by following the procedure in Ref. 12. We start with T ^ / C ^ ^ ) } =
T r
{r
5
/ ( ^ ^ ) } = n+ - n_
(29)
Namely, the index is not modified by any regulator f(x) with /(0) = 1 and f(x) rapidly going to zero for x —¥ oo, as can be confirmed by using (2.14). This means that you can use any suitable f(x) in the evaluation of the index by taking advantage of this property. We then consider a local version of the index tT{r5f(^^)}(x,X)=tr{(75-H^1)f(^^)}(x,x)
(30)
where trace stands for Dirac and Yang-Mills indices; Tr in (3.7) includes a sum over the lattice points x. A local version of the index is not sensitive to the precise boundary condition , and one may take an infinite volume limit of the lattice in the above expression. 83
1936
K. Fujikawa
We now examine the continuum limit a —> 0 of the above local expression (3.8) a . We first observe that the term tr{ff2fc+i/((2^)!)}
(31)
goes to zero in this limit. The large eigenvalues of H = a^D are truncated at the value ~ aM by the regulator f(x) which rapidly goes to zero for large x. In other words, the global index of the operator T*H2k+1f(&jJ°)l) ~ 0(aM)2k+1 -> 0 for a ->• 0 with fixed M. We thus examine the small a limit of tr{75/(%^)}-
(32)
The operator appearing in this expression is well regularized by the function f(x), and we evaluate the above trace by using the plane wave basis to extract an explicit gauge field dependence. We consider a square lattice where the momentum is defined in the Brillouin zone _
7T
37T
W ^ ) } - ^ )
4
/ ^ ) ^
(34)
for a —> 0 if one chooses f(x) = e~x, for example. We thus examine the above trace in the momentum range of the physical species
"5
s
<35)
•*•<£•
We obtain the limiting a —> 0 expression
Js'/II^™^
Akx
/ ..
+
fL
dk d4k „ _ * , ,n 4.Ae~tkx~(5f(V
_L (27r)
dAk
_ikx
Al,D)\ M2
.Ai-ysP)2,
ikx
L—>oo 7,2
= tr{75/(^)}
(36)
a
T h i s continuum limit corresponds to the so-called "naive" continuum limit in the context of lattice gauge theory. 84
Generalized Ginsparg-Wilson Algebra and Index Theorem 1937
where we first take the limit a —> 0 withfixedfcMin — L oo. This procedure is justified if the integral is well convergent.12 We also assumed that the operator D satisfies the following relation in the limit a —» 0 Deikxh(x)
->• eikx{-
ty+ifi-g4)h(x)
= i ( 0 + i 5 4)(eifca;/i(x)) = * ^ ( e ^ a : ) )
(37)
for anyfixedfc^,(—5^ < k^ < 5^), and a sufficiently smooth function h(x). The function /i(a;) corresponds to the gauge potential in our case, which in turn means that the gauge potential A^x) is assumed to vary very little over the distances of the elementary lattice spacing. Our final expression (3.14) in the limit M -> 00 reproduces the Pontryagin number in the continuum formulation (with e1234 = l ) 5 Jim^ tr75/GP2/M2) = t r y s ^ l i y , l ^ f
j
^-J"{-k^)
= ^ t r e ^ F^Fap.
(38)
When one combines (3.7) and (3.16), one reproduces the Atiyah-Singer index theorem (in continuum R4 space). 7 ' 8 We note that a local version of the index (anomaly) is valid for Abelian theory also. The global index (3.7) as well as a local version of the index (3.8) are both independent of the regulator / ( # ) provided5 /(0) = 1,
/(oo)=0,
f'(x)x\x=0=f'(x)x\x=oo=0.
(39)
We have thus established that the lattice index in (3.7) for any algebraic relation in (1.2) is related to the Pontryagin index in a smooth continuum limit as n+-n^=J d'x^tve^^F^F^.
(40)
This shows that the instanton-related topological property is identical for all the algebraic relations in (1.2), and the Jacobian factor (3.6) in fact contains the correct chiral anomaly. (We are implicitly assuming that the index (3.7) does not change in the process of taking a continuum limit.) A detailed perturbative analysis of chiral anomaly for the general operators with k > 0 has been performed, and the above result has been confirmed.12 Also a numerical study of the index relation has been performed: The numerical result indicates the consistency of our analyses. 13 4. Explicit construction of the lattice Dirac operator for k > 1 We now comment on an explicit construction of the lattice Dirac operator which satisfies the generalized algebraic relation (1.2) with k > 0. We start with the conventional Wilson fermion operator Dw defined by Dw(x, y) = i-y^C^x, y) + B(x,y) 85
-m0SXty,
1938 K. Fujikawa
C»(x,y) = -z-iSx+favUpiy) 2a B(x, y) = — J2i2S^y
Sx,y+liaUl(x)],
~ sy+Ha,xUl(x) - (5V)X+AaC/M(y)],
U^y) = exp[iagAp(y)],
(41)
where we added a constant mass term to Dw for later convenience. The parameter r stands for the Wilson parameter. Our matrix convention is that 7^ are antihermitian, (7**)* = —7M, and thus (J! = 7MCM(n, m) is hermitian
(42)
The Dirac operator for a general value of k is constructed by rewriting (1.2) as a set of relations H2k+1l5
+ l5H2k+1
2H2^2k+1\
=
# 2 7 s - 7 5 # 2 = 0,
(43)
with H — a^D. The second relation in (4.3) is shown by using the defining relation (1.4), and the first of these relations (4.3) becomes identical to the ordinary Ginsparg-Wilson relation (1.1) if one defines H^k+i) = H2k+1. One can thus construct a solution to (4.3) by following the prescription used by Neuberger 2 H(2k+1) = ll5[l + Dwk+1)— 2
* 1 / m w( 2k^)W f c +wl kU t n ( 2 f c + l ) y/(D
(44)
where Dwk+1)
= i((?)2k+1 + B2k+1
- (^°)2*+i
(45)
The operator H itself is then finally defined by (in the representation where H^k+i) is diagonal) H = {H(2k+1)y'2k+l
(46)
in such a manner that the second relation of (4.3) is satisfied. This condition (4.3) is shown to be satisfied in the representation where H^k+i) ls diagonal. 12 Also the conditions 0 < mo < 2r = 2 and 2m 2fc+1 = 1
(47)
ensure the absence of species doublers and a proper normalization of the Dirac operator H. 5. Locality p r o p e r t i e s of general o p e r a t o r s We have explained that the general operators for any finite k give rise to correct chiral anomaly and index relations in the (naive) continuum limit. This suggests 86
Generalized
Ginsparg-Wilson
Algebra and Index Theorem
1939
that those operators are local for sufficiently smooth background gauge field configurations. The locality of the standard overlap operator with k = 0 has been established by Hernandez, Jansen and Liischer,15 and by Neuberger. 16 As for the direct proof of locality of the operator D for general k, one can show it for the vanishing gauge field by using the explicit solution for the operator H in momentum representation 12 ' 13 H(ap,)=l5(h^(^=)^{(J^ ^ \/H,,,
+
v
Mk)^-(JH^-M k)^r^} v a
(48) where FW = (s2fk+1
+ Ml
Mfc = E ( l - c M ) ] 2 f c + 1 - m 2 f e + 1
(49)
and s,j, = smapfj, Cf,, — cos ap^ . / = 7 M sinap M .
(50) 2
For k = 0, this operator is reduced to Neuberger's overlap operator. Here the inner product is defined to be s 2 > 0. This operator is shown to be free of species doublers for the parameter mo within the range 0 < mo < 2 when we set r — 1, and 2m0 = 1 gives a proper normalization of H, namely, for an infinitesimal pM, i.e., for \apfjt]
(51)
to be consistent with H = 75a!?; the last term in the right-hand side is the leading term of chiral symmetry breaking terms. The locality of this explicit construction (5.1) is shown by studying the analytic properties in the Brillouin zone. 12 It is important to recognize that this operator is not ultra-local but exponentially local; 17 the operator H{x, y) decays exponentially for large separation in coordinate representation H(x, y) ~ exp[-|a: - y\/(2.5ka)}.
(52) n
An explicit analysis of the locality of the operator H(2k+i) ( °t H itself)in the presence of gauge field, in particular, the locality domain for the gauge field strength | |FM„11 has been performed. The locality domain for | {F^11 becomes smaller for larger k, but a definite non-zero domain has been established. 12 The remaining task is to show the locality domain of \\Fpv\\ for the operator H = (H^k+i))1//'2fc+1). Due to the operation of taking the {2k + l)th root, an explicit analysis has not been performed yet, though a supporting argument has been given. 12 87
1940 K. Fujikawa 6. Conclusion We have reported the recent investigation of topological properties of a general class of lattice Dirac operators denned by the algebraic relation (1.2). All these operators satisfy the index theorem and thus they are topologically proper. A precise proof of the locality of these general Dirac operators with fully dynamical gauge fields remains to be formulated. The operators with large k is expected to exhibit infrared singularities in perturbative analyses as is suggested by the construction of H^k+i) in (4.4), and thus the Wilsonian formulation of effective action, which is supposed to be free of infrared singularities, would be essential. Although we discussed only 4-dimensional theory, the recent developments in the treatment of lattice fermions11 may have some implications on 2-dimensional theory also, which is the main subject of this Symposium. In this respect, the fact that the lattice Dirac operators are not ultra-local but exponentially local 15 may be of some interest. See Ref. 18 for a Ginsparg-Wilson construction on a 2-dimensional fuzzy sphere. Acknowledgments I thank Mo-lin Ge for the hospitality at Nankai University. References 1. P.H. Ginsparg and K.G. Wilson, Phys. Rev. D25 (1982) 2649. 2. H. Neuberger, Phys. Lett. B417 (1998) 141; B427 (1998) 353. 3. P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. Lett. B427 (1998) 125, and references therein. 4. M. Liischer, Phys. Lett. B428 (1998) 342. 5. K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195; Phys. Rev. D21 (1980) 2848; D22 (1980) 1499(E). 6. Y. Kikukawa and A. Yamada, Phys. Lett. B448 (1999) 265. D.H. Adams, "Axial anomaly and topological charge in lattice gauge theory with overlap-Dirac", hep-lat/9812003. H. Suzuki, Prog. Theor. Phys. 102 (1999) 141. K. Fujikawa, Nucl. Phys. B546 (1999) 480. 7. R. Jackiw and C. Rebbi, Phys. Rev. D16 (1977) 1052. 8. M. Ativan, R. Bott, and V. Patodi, Invent. Math. 19 (1973) 279. 9. As for reviews of chiral anomaly in continuum theory, S.L. Adler, in Lectures on Elementary Particles and Quantum Field Theory, edited by S. Deser et al. (MIT Press, Cambridge, Mass., 1970). S.B. Treiman, R. Jackiw, B. Zumino and E. Witten, Current Algebra and Anomalies (World Scientific, Singapore, 1985). R. Bertlmann, Anomalies in Quantum Field Theory (Oxford University Press, Oxford, 1996). K. Fujikawa, "Aspects of anomalies in field theory", Int. J. Mod. Phys. A16 (2001) 331. 10. T.W. Chiu, Phys. Rev. D58 (1998) 074511. 11. As for reviews of recent developments in lattice fermions, F. Niedermayer, Nucl. Phys. Proc. Suppl. 73 (1999) 105. 88
Generalized Ginsparg-Wilson Algebra and Index Theorem
12.
13. 14. 15. 16. 17. 18.
1941
H. Neuberger, "Exact chiral symmetry on the lattice," hep-lat/0101006. M. Liischer, "Chiral gauge theories revisited," hep-th/0102028. K. Fujikawa, Nucl. Phys. B589 (2000) 487. K. Fujikawa and M. Ishibashi, Nucl. Phys. B587 (2000) 419. K. Fujikawa and M. Ishibashi, Nucl. Phys. B605 (2001) 365. T.W. Chiu, Nucl. Phys. B588 (2000) 400; Phys. Lett. 4 9 8 B (2001) 111; Nucl. Phys. Proc. Suppl. 94 (2001) 733. K. Fujikawa, Phys. Rev. D60 (1999) 074505. P. Hernandez, K. Jansen and M. Liischer, Nucl. Phys. B552 (1999) 363. H. Neuberger, Phys. Rev. D61 (2000) 085015. I. Horvath, Phys. Rev. Lett. 81 (1998) 4063. A. P. Balachandran, T. R. Govindarajan, and B. Ydri, hep-th/0006216.
89
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1943-1950 © World Scientific Publishing Company
S P E C T R A L F L O W OF HERMITIAN W I L S O N - D I R A C OPERATOR A N D T H E I N D E X THEOREM IN ABELIAN G A U G E THEORIES O N FINITE LATTICES
T. FUJIWARA Department
of Mathematical
Sciences, Ibaraki University, Mito 310-8512,
Japan
Received 15 November 2001 Final Form 17 January 2002 The spectral flows of the hermitian Wilson-Dirac operator for a continuous family of abelian gauge fields connecting different topological sectors are shown to have a characteristic structure leading to the lattice index theorem. The index of the overlap Dirac operator is shown to coincide with the topological charge for a wide class of gauge field configurations. It is also argued that in two dimensions the eigenvalue spectra for some special but nontrivial background gauge fields can be described by a set of universal polynomials and the index can be found exactly.
1. Introduction For a long time it has been considered that chiral symmetries cannot be implemented on the lattice due to the Nielsen-Ninomiya no-go theorem. The situation, however, has completely changed after the discoveries of lattice Dirac operators satisfying the Ginsparg-Wilson (GW) relation. 1_3a It is now possible to define exact chiral symmetry on the lattice. 6 Furthermore, the index theorem on the lattice relating the index of the GW Dirac operator to chiral anomaly has been obtained by Hasenfratz, Laliena and Niedermayer (HLN). 6 ' 7 In continuum theories the nontrivial topological structure of gauge fields are considered to be responsible for nonperturbative phenomena such as the large 77-77' mass splitting in QCD and the fermion number violation in the standard model. The Atiyah-Singer (AS) index theorem provides a key relation there. The HLN index theorem is known to reproduces the AS index theorem in the classical continuum limit.8 However, it is not so clear what topological structure of the space of lattice gauge fields is related to the index of the GW Dirac operator on finite lattices. We expect that an extension of the index theorem relating the index directly to the topological invariants of gauge fields can be established also on finite lattices. a
For reviews, see Niedermeyer 4 and Liischer. 5 91
1944
T. Pujiwara
In this talk we would like to propose the "index theorem" relating the index of the GW Dirac operator directly to topological invariants of the lattice gauge fields,9-12 mostly based on our recent results. 13 Working with the overlap Dirac operator 3 for compact U(l) theories on finite periodic lattices in two and four dimensions, we investigate the spectrum of the hermitian Wilson-Dirac operators numerically for a family of link variables connecting constant magnetic field configurations with distinct topological charges. Such an analysis has already been carried out by Narayanan and Neuberger14 within the overlap formalism. They found the characteristic structure of the spectral flows leading to the index theorem. We will extend their analysis to include strong and nonsmooth gauge fields and find that the "index theorem" is kept intact for a wider class of gauge fields than those expected from the locality bounds. 15 ' 16 The index and the topological charge, however, behave completely differently for nonsmooth gauge fields and the coincidence between the index and the topological charge breaks down. We also argue that the characteristic structure of the spectrum of the hermitian Wilson-Dirac operator in two dimensions can be understood exactly and the index can be computed for some special gauge field configurations. 2. The index theorem on the lattice Let us begin with the definition of the overlap Dirac operator. On a d = 2N dimensional euclidean hypercubic regular lattice it is defined by
^= 1+7d+17P'
(1)
where H is the hermitian Wilson-Dirac operator given by H1>{x) =
7d+
i | (d- m)iP(x) - JT (izJitu^xMx
+ A)
+^^(*-AM*-£)]}•
(2)
We have chosen the lattice spacing a = 1 and the Wilson parameter r = 1. The link variables U^x) and the fermion wave function ip(x) are assumed to be periodic. The 7-matrices are taken to be hermitian and 7d+i = (—1)271 •••'yd is employed. The m is chosen in the range 0 < m < 2 to avoid species doubling and is taken to be m — 1 in our analysis. Obviously, D is only well-defined for the gauge field configurations with det H ^ 0. It may exhibit discontinuities at the link variables for which H has zero-modes. This can be seen by noting the relation between the index of D and the spectral asymmetry 17 of H indexD = Tr 7 d + 1 ( 1 - \D\
= " ^ T r - ^ L = = -N+~N~ 92
,
(3)
Spectral Flow of Hermitian Wilson-Dirac Operator 1945
where N+ (JV_) is the number of positive (negative) eigenvalues of H. Any two gauge field configurations with distinct indices cannot be continuously deformed into each other without crossing the configurations with detH = 0. By excising such singular gauge field configurations the space of the link variables becomes disconnected. The HLN index theorem associates indexD to each connected component of the space of link variables. Another way to give nontrivial topological structure to the space of link variables is to impose the conditions 9 sup||l-PMI,(x)||<77,
(4)
where P^ is the standard plaquette variable and r] (< 2) is a positive constant. The space of link variables becomes disconnected for sufficiently small 7/ and it is possible to assign topological charge to each connected component. In abelian theories the explicit forms of the topological charge 12 ' 18 is given by
x • • • x FliNVN (x + /ii + v\ H
h /tjv-i + i>N-i) ,
(5)
where F^v = —ihiP M „ (FliU(x)\ < TT) is the field strength and eMl...Md is the LeviCivita symbol in d = 2N dimensions. The QN is a smooth function of the link variables within a connected component and takes an integer value given by 10 ' 12
where 27rmM^ is the magnetic flux through /xi/-plane. In fact it can be shown that the space of link variables of abelian gauge theory is decomposed into a finite number of connected components characterized by a set of integers { m p } , and any two configurations with the same set of magnetic fluxes can be continuously deformed into each other without violating (4). In general one can find gauge field configurations that satisfy (4) and det H = 0 simultaneously. If it happens, the index£) may jump within a connected component satisfying (4). However, if we choose n to satisfy
0<7?<
<7)
|z=f'
H cannot have zero-modes 15,16 and the indexD is a constant in each connected components satisfying (4). It is very natural to expect that the index and the topological charge coincide with each other. The precise form of the "index theorem" for abelian gauge theories can be stated as : For the link variables satisfying (4) and (7) the index of D and the topological charge QN are related by indexD = (-l)NQN 93
.
(8)
1946
T. Fujiwa.ro.
-15
-10
-5
0
5
10
15
Parameter t
Fig. 1.
Eigenvalue spectrum of H for L = 6 in two dimensions.
One way to find indexD is to count the net number of spectral flows of H as m varies from 0 to 1 while keeping the link variables. This approach was adopted by several authors. 14 ' 19 ' 20 To see how the index of D behaves as one varies the link variables continuously from one connected component to another it is more convenient to investigate spectral flows of H for a continuous family of link variables connecting configurations with constant field strengths Ftiv(x) = 2irmflv/L2. Concretely, we take the one-parameter family of link variables in two dimensions
U?\x)=exp
-lt —
X20XuL-l
r(0 (x) — exp U2
.
2TT
(0 < zM < L) (9)
The field strength is a constant for any value of t except for the corner point %\ 2 = L — 1. As a function of t, the topological charge Q\ has discontinuities at ' (2j + l ) i 2 2(Z,2_i) •j + \(j : 0 , ± 1 , ± 2 , - " ) . 3. Numerical results in two and four dimensions We have numerically analyzed the eigenvalue spectrum of H over the range —L 2 /2 < t < L2/2 for 2 < L < 15. It is helpful to note the following facts: (1) In two dimensions the eigenvalues A of H are bounded by |A| < 3. 1 6 (2) The eigenvalue spectrum of H is L2 periodic in t. (3) The eigenvalue spectrum of H is symmetric with respect to the point t = 0. Hence indexD is an L2 periodic odd function of t and vanishes at t = 0, ± L2/2, where the spectrum is symmetric. In Figure 1 the whole spectrum of H is shown for L = 6. It is consistent with the result obtained by Narayanan and neuberger. 14 The characteristic structure of the spectrum is not changed with L. We find that most of the eigenvalues lie within the upper and the lower trapezoid regions symmetrically separated by the parallelogram region and there are large gaps at integer t. As t increases by unity from an integer, an eigenvalue belonging to the lower trapezoid crosses the parallelogram upward 94
Spectral Flow of Hermitian Wilson-Dirac Operator 1947
4 2 0 Q
"2
X
-o
-4
c -6 -8 -10
-12 0
2
4
6
8
10
12
Parameter t Fig. 2. The indexD and - Q i are plotted for 0 < t < L2/2 (L = 5, d = 2).
a n d moves to the upper trapezoid. In particular on the interval of the horizontal axis c u t by the parallelogram eigenvalues change the sign for a sequence of the values of t, where the index j u m p s by —1. T h e i n d e x D and —Qi a r e plotted over the region 0 < t < L2/2 for L = 5 in F i g u r e 2. T h e correspondence between the index and the topological charge is very excellent for 0 < t < L2/4 except for the points around the discontinuities. This is consistent with the numerical results given by Chiu. 2 1 F o r t > L? /A the index£) a n d —Qi behaves completely differently. Such can b e considered as a kind of lattice specific phenomena. It is safer t o avoid such configurations in order t o keep proper connection with continuum theories. W e have also carried o u t a similar analysis in four dimensions by taking the link variables (9) in the presence of a constant magnetic flux t h r o u g h 34-plane. T h e characteristic features of t h e spectral flows observed in two dimensions can be seen also in four dimensions. In Figure 3 we indicate the spectrum of i f for L = 4, |A| < 1, 77134 = 1 and \t\ < 8. A well-isolated eigenvalue crosses t h e horizontal axis, w h e r e index£> increases by one unit. The characteristic feature of t h e spectrum is not changed for 77234 > 1. I n general 77234 adjacent eigenvalues flow downward and i n d e x D increases by 77134 w h e n they cross the i-axis. This is consistent with (6). T h o u g h our analysis is restricted to rather small lattice sizes 2 < L < 4, we a n t i c i p a t e t h a t the parallelogram region expands rapidly enough as L increase and t h e coincidence between t h e index and the topological charge occurs for a wide class of gauge field configurations. Incidentally, for gauge fields satisfying (4) and (7) nonvanishing topological charges can be realized only for L > 9. 4. S o m e exact results in t w o dimensions C o m i n g back to two dimensions, we now show that the characteristic structure of t h e s p e c t r u m of H can b e u n d e r s t o o d by noting that the eigenvalue equations for c o n s t a n t field s t r e n g t h configurations can b e converted to equivalent simple one95
1948
T. Fujiwara
\\J^%
Ill': 0.5
SSfwiSS& . ! $ & < -
6
-
\ilfi
4
-
2
0
2
4
6
8
Parameter t
Fig. 3. Eigenvalue spectrum of H for |A| < 1, s = 1 and |t| < 8 (L — 4 and d = 4). dimensional systems. Here we will consider the cases of field configurations at t = r and t = L2/r = sL, where r and s sue arbitrary positive integers satisfying L = rs. At t = sL the H is independent of X2 and r periodic in xi. This implies that the system can be converted to a smaller one with degrees of freedom 2r by Fourier transformations. It can be shown that the eigenvalues A of H at t — sL satisfy the secular equation det
B(p,q)-X C(p,q)i
C(p,q) -B(p,q)-X
(10)
= 0
where p and q are the Fourier momenta P
2irk — '
2rirl q
=—
(0 < fc < L ,
'
0 < Z < s)
(11)
and B(p,q), C(p,q) are r x r matrices defined by (B(p,q))kl
= --6™+1
+ 1 - cos p +
2-wk °k,l
9°fc+l,Z '
(?)
fc+l,i
(0
(12)
s(«)i ;is the Kronecker's rJ-symbol for 0 < k, I < r — 1 and satisfies the twisted The (5^. boundary conditions 6^'r = e~ig5k,o and <S^ = etq5o,kThe secular equation (10) can be rewritten in the following form , ,.,
frW=
(_l)r-l . -, 2r'_4
. 2rP
• 2Q
sin2^sm2| ,
(13)
where / r = A2r + • • • is a polynomial of degree 2r and is independent of p and fj. For 1 < r < 4 it is explicitly given by A = A2-1, 96
Spectral Flow of Hermitian
Wilson-Dime
Operator
1949
/ 2 = A4 - 6A2 + 1 , /3 =
A6-9A4 + ^ A 2 - 3 V 3 A - i ,
U = A8 - 12A6 + 42A4 - 8A3 - 44A2 + 24A - 3 .
(14)
In particular r = 1 corresponds to the free spectrum since the link variables become trivial. The eigenvalues A must satisfy the inequality 0 < (-l)r_12r-4/r(A) < 1 .
(15)
This gives rise to 2r allowed narrow intervals for A. In each interval there appears exactly Lxs eigenvalues. As an example, we consider the case that L = 4s and take r = 4. For any p, q the eight roots of (13) are separately located in well-separated eight narrow intervals. There are three negative eigenvalues and five positive eigenvalues for each p, q. Then the index is then given by— 1 x L x s = —sL. This coincides with —Q\. In general fr{\) = 0 has r — 1 negative roots and r +1 positive roots for r > 4 and, hence, indexD — —sL — —Q\ at t = sL. On the other hand it is vanishing for 1 < r < 3 since the number of positive roots and that of negative ones are always equal. It is possible to extend the above arguments to integers t = r (< L). In particular we arrive at the relation det(fl--A)|t = T . = ( / , L ( A ) ) p .
(16)
This gives at t = r the index — r = —Q\ for sL > 4 and 0 for sL < 3. These are completely consistent with the numerical results. 5. Summary We have confirmed the equality (8) between the index and the topological charge for abelian gauge theories on finite periodic lattices in two and four dimensions. It holds true for a wider class of gauge field configurations than those satisfying the locality bounds. 15 ' 16 The condition (4) with (7) excludes uniformly the configurations for which the discrepancy between the index and the topological charge appears, ensuring the index theorem (8). References 1. 2. 3. 4. 5. 6. 7. 8.
P.H. Ginsparg and K.G. Wilson, Phys. Rev. D25, 2649 (1982). P. Hasenfratz, Nucl. Phys. (Proc. Suppl.) 63, 53 (1998); Nucl. Phys. B525 401 (1998). H. Neuberger, Phys. Lett. B417, 141 (1998); B427, 353 (1998). F. Niedermayer, Nucl. Phys. B (Proc. Suppl.) 73, 105 (1999). M. Liischer, CERN-TH/2001-031. M. Liischer, Phys. Lett. B428, 342 (1998). P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. Lett. B427, 125 (1998). Y. Kikukawa and A. Yamada, Phys. Lett. B448, 265 (1999); D.H. Adams, hep-lat/9812003; H. Suzuki, Prog. Theor. Phys. 102, 141 (1999). 97
1950
T. Pujiwara
9. M. Liischer, Commun. Math. Phys. 85, 39 (1982). 10. A. Phillips, Ann. Phys. 161, 399 (1985). 11. M. Gockeler, A.S. Kronfeld, M.L. Laursen, G. Schierholz and U.-J. Wiese, Nucl. Phys. B 2 9 2 , 349 (1987). 12. T. Fujiwara, H. Suzuki and K. Wu, Prog. Theor. Phys. 105, 789 (2001). 13. T. Fujiwara, IU-MSTP/44, hep-lat/0012007. 14. R. Narayanan and H. Neuberger, Phys. Rev. Lett. 7 1 , 3251 (1993); Nucl. Phys. B443, 305 (1995). 15. P. Hernandez, K. Jansen and M. Liischer, Nucl. Phys. B 5 5 2 , 363 (1999). 16. H. Neuberger, Phys. Rev. D 6 1 , 085015 (2000). 17. R. Narayanan and H. Neuberger, Nucl. Phys. B412, 574 (1994). 18. M. Liischer, Nucl. Phys. B538, 515 (1999); T. Fujiwara, H. Suzuki and K. Wu, Phys. Lett. B463, 63 (1999); Nucl. Phys. B569, 643 (2000); IU-MSTP/37, hep-lat/9910030. 19. S. Itoh, Y. Iwasaki and T. Yoshie, Phys. Rev. D36, 527 (1987). 20. R. Edwards, U. Heller and R. Narayanan, Nucl. Phys. B 5 2 2 , 285 (1998). 21. T.W. Chiu, hep-lat/9911010.
98
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1951-1961 © World Scientific Publishing Company
DIMERS A N D S P A N N I N G TREES: SOME R E C E N T RESULTS
F. y. wu Department of Physics, Northeastern University Boston, Massachusetts 02115, U. S. A.
Received 4 December 2001 This paper reviews some recent progress on dimer and spanning tree enumerations. We use the Kasteleyn formulation to enumerate close-packed dimers on a simple-quartic net embedded on non-orientable surfaces, and obtain solutions in t h e form of double products. For spanning trees the enumeration is carried out by evaluating t h e eigenvalues of the Laplacian matrix associated with the lattice, a procedure which holds in any spatial dimension. In two dimensions a bijection due to Temperley relates spanning tree and dimer configurations on two related lattices. We use this bijection to enumerate dimers on a net with a vacancy on the boundary. It is found t h a t t h e occurrence of a vacancy induces a VJV correction to the enumeration, where N is t h e linear size of t h e lattice, and changes the central charge from c = 1 to —2.
1. Introduction Theoretical studies of the physics of real systems often lead to problems of farreaching interests in mathematics, and solutions to the mathematical problems in turn yield new insights to the physical problems. One such example is the advent of dimer statistics, a subject matter at the forefront of mathematical research, from the evaluation of the adsorption entropy of diatomic molecules on a surface. 1 Another example is the arising of the notion of spanning trees, again a subject matter of immense interest in graph theory, from the theory of electric network currents. 2 In two dimensions these two mathematical problems are further interrelated, a fact recognized again through the consideration of the physics of the problems. 3 In this paper we describe and review some recent progress on dimers and spanning trees obtained by the author and co-workers.4-8 We first define the problems of dimer and spanning tree enumerations. The dimer problem is treated in section 2 where the Kasteleyn formulation is outlined and used to obtain the dimer generating function for two non-orientable surfaces, the Mobius strip and the Klein bottle. In section 3 an established result in graph theory is used to enumerate spanning trees. In section 4 we describe a bijection due to Temperley3 which relates dimers and spanning trees on two related lattices, and use it to establish the independence of the dimer generating function on the location of a vacancy on the boundary of a simple-quartic net. The bijection also 99
1952 F. Y. Wu
leads to an explicit evaluation of the dimer generating function for the defect lattice. Results of finite-size analyses, which lead to corrections induced by the geometry of a boundary vacancy, are presented in section 5. 2. Definitions We shall consider a regular lattice C having a vertex (site) set V and edge set E, but much of our results are applicable more generally to C being an arbitrary graph. Number the sites from 1 to |V| and associate to the edge ejj connecting vertices i and j a weight x^, with x^ = 0 if there is no edge connecting i and j . A dimer covering V (for |V| = even) is a pairing of the |V| vertices into |V|/2 pairs. We say that the edge eij is covered by a dimer if ejj appears in V. Then, the dimer generating function is
Z(C;{xij}) = J2 I ] *«'
(^
where the summation is taken over all dimer coverings. The total number of dimer coverings is obtained by setting x^ = 1, or Ndimer(C)
= Z(C; 1).
(2)
Specializing (1) to an M. x M simple-quartic lattice of M. rows and Af columns with edge weights Zh and zv respectively in the horizontal and vertical directions, the dimer generating function is
ZM,Mzh,zv) = J24h^v,
(3)
V
where n^ and nv are, respectively, the numbers of horizontal and vertical dimers in V. Next we define spanning trees. A subset of edges T C E is a spanning tree if it has \V\ — 1 edges with at least one edge incident at each vertex. Thus T has no cycles. The enumeration of spanning trees concerns with the evaluation of the spanning tree generating function
I I x*>
T(C;{Xij})=J2
(4)
TCE eijCT
where the summation is taken over all spanning trees configurations T. Particularly, the total number of spanning trees on £ is obtained by setting x^ = 1, or JVgPT(£) = T(£; 1).
(5)
Specializing (5) to a simple-quartic net as in the above, the tree generating function assumes the form
rMAK£;*fc,*«) = X > r C > T 100
(6)
Dimers and Spanning Trees: Some Recent Results
1953
where n/, and nv are, respectively, the numbers of horizontal and vertical edges in T. The evaluation of (3) for £ with free and toroidal boundary conditions was first accomplished by Kasteleyn 9 and Temperley and Fisher. 10 ' 11 Here we extend the solution to non-orientable surfaces, 4 ' 7 ' 12 and to the net £ having a boundary vacancy. 8 3. Dimer enumerations 3.1. Kasteleyn
formulation
It is an elementary fact that the superposition of two dimer configurations decomposes a lattice £ into superposition polygons, namely, polygons formed by tracing along dimers from vertex to vertex. Orient all edges of £ and define an |V| x \V\ antisymmetric matrix A(xij) with elements Aij — —Aji = Xij,
if the edge ij is directed from i to j .
(7)
Then, Kasteleyn 9 has established the remarkable result that Z(C;{xij})
= y/\A(xij)\,
(8)
where |A| is the determinant of A, provided that lattice edges are oriented such that the product of all edge weights around every possible superposition polygon is negative. Namely, XijXjk • • • xu < 0
(the Kasteleyn criterion)
(9)
for sites i,j, k, ...,£ around a superposition polygon arranged in the order of, say, a clockwise (cw) direction. Henceforth we shall refer to the sign of —XijXjk •••xu as the sign of the (superposition) polygon. The Kasteleyn criterion (9) is remarkable since it says nothing about the dimensionality of the lattice. It is this flexibility which permits its application to non-orientable surfaces. However, even if the Kasteleyn criterion is met, it still remains to evaluate the determinant |A(xy)| which can be a formidable task in some cases. 3.2. Simple-quartic
nets on non-orientable
surfaces
We consider the enumeration of dimers on an M x H simple-quartic net embedded on two non-orientable surfaces. The net forms a Mobius strip if there is a twisted (Mobius) boundary condition in the horizontal direction as shown in Fig. 1, and a Klein bottle if, in addition to the twisted boundary condition, there is also a periodic boundary condition in the vertical direction. To satisfy the Kasteleyn criterion (9) we make use of a trick due to T. T. Wu 13 of associating a factor i to dimer weights in one spatial direction. For a simplequartic net with free boundaries, one associates a factor i to dimer weights in the 101
1954
F. Y. W%
A
•*—
B
-«— C
C
•+—B
D
-<-
A
-*— E
B
-«— D
C
-«— C
D
-+—
E
-«- A
D
A
(a)
B
(b)
Fig. 1. An MxM Mobius strip and the associated edge orientations. A, B, C, D, E are repeated sites, (a) (M,N) = (5,4). (b) (M,N) = (4,5).
direction in which the number of lattice sites is odd. For {M,Af} = {even, odd}, for example, one replaces ZH by izn- If the number of lattice sites is even in both directions, then the factor i can be associated to dimers in either direction. Next one orients all parallel lattice edges uniformly in the same direction. To see that this orientation satisfies the Kasteleyn criterion for free boundary conditions, the case considered by Wu, one superimposes any dimer covering C\ with a standard Co in which the lattice is covered only by parallel dimers with real weights. Then, each superposition polygon formed by C\ and Co contains an even number of arrows pointing in each (cw or ccw) direction as well as a factor i4n+2 = — 1, where n is a nonnegative integer. It follows that the criterion (9) holds and the sign of every polygon is positive. Note that the construction of Co requires either M. or J\f to be even. On non-orientable surfaces a superposition polygon can wrap around the lattice in the horizontal direction, and this causes problems in realizing the Kasteleyn criterion. Particularly, one needs to pay attention to whether M. and M are even or odd. Starting from a net with free boundaries, there are M additional horizontal "connecting" edges (shown in Fig. 1) and, in the case of the Klein bottle, M additional vertical connecting edges (not shown in Fig. 1). These edges need to be oriented. It turns out that, except in the case that both M,Af are even, 4 it is not possible to orient the connecting edges so that the Kasteleyn criterion holds for all polygons wrapping around the lattice. However, one can take advantage of the regularity of the signs of the polygons to extract the desired solution. Let the horizontal connecting edges carry a weight z, and let A(z) denote the resulting antisymmetric matrix (7). It is straightforward to show 7 that superposition polygons containing 4n and 4n + 1 connecting z edges in its perimeter, where n is an integer, have the same sign and those having An+ 2 and 4n + 3 connecting z edges have the opposite sign. It follows that 7 the desired dimer generating function 102
Dimers and Spanning Trees: Some Recent Results 1955 on a net (with uniform weights ZH a n d zv) is given by the linear combination
(1 - i)y/\A(izh)\
+ (1 +
i)y/\A(-izh)
R e (l-i)y/\A(iZH)\
(10)
where Re denotes t h e real p a r t . This formula applies to b o t h the Mobius strip and the Klein bottle. T h e determinant \A(±izh)\ can b e evaluated by computing the eigenvalues of the matrix A(±izh), a n d t h e algebra is somewhat different for the two lattices shown in Fig. 1. We refer t o Ref. 7 for details a n d give here only the final result,
MAT/2 D z. Re
r
[{M+i)/2\ H
(
a-o n nh-i)* m=\
+m+i
n=l
SA
,\
S in^^+2x m (11)
where [x] is the integral p a r t of x, a n d Xm =
— Zh^
cos •
for t h e Moebius strip
M +l
(2m - 1)TT
Zh
sin
for t h e Klein bottle.
M
(12)
For N = even, t h e p r o d u c t in (11) is real a n d (11) reduces to a simpler form r,
ir
\
MM/2
n n h ^ ^ ^ . as)
2N m=l n=l For the Mobius strip Tesler 1 2 has also obtained t h e solution in terms of generalized Fibonacci numbers. It c a n b e shown t h a t his solutions are the same as those given AM,SS{L;
Zh, zv)
=
zh
by (11). In all cases, one o b t a i n s in t h e t h e r m o d y n a m i c limit the per-site bulk free energy 1 Jdimer\Zh, Zv)
In
M,AT-+°o MM 1 r^ f^
ZM,u{£; 4 z\ sin 2 6 + 4 zl sin 2
(14)
This gives the per-site e n t r o p y of t h e adsorption of diatomic molecules on a simplequartic lattice as G JdimerV-'-; 1)
(15)
where G is the C a t a l a n c o n s t a n t given by G = 1 - 3 - 2 + 5 ~ 2 - 7~2 + • • • = 0.915 965 594. 103
(16)
1956
F. Y. Wu
4. Spanning tree enumerations 4.1. The Laplacian
matrix
First we recall an established result in graph theory on spanning trees. Consider a graph £ with edge weights x^. In analogous to (7) we define a |V| x |V| symmetric matrix B with elements Bij = Bji = Xij = 0
if i and j are adjacent otherwise.
(17)
Thus B is the adjacency matrix of C if x^ = 1. Define further a diagonal matrix A with elements
Au = J2xiJ-
(18)
Q= A - B
(19)
Then, the matrix
is the Laplacian of the lattice C. The Laplacian matrix has the property that the sum of each row or column is zero, so it has a zero eigenvalue. Let Ai, A2,..., A|v|-i be the |V| - 1 nonzero eigenvalues of £. Two fundamental theorems in graph theory 14,15 state that we have T(C\ {Xij}) = any cofactor of Q 1
(20)
|v|-i
A proof of (20) can be found in any book of graph theory (see, e.g., Ref. 14), and an elementary proof of the equivalence of (20) and (21) has been given in Ref. 5.
4.2. Simple-quartic
lattice
We have used the formulation (21) to derive spanning tree generating functions for finite hypercubic lattices in d dimensions under various boundary conditions 5 as well as for regular lattices in two dimensions.6 Here we discuss only the simple-quartic lattice with free boundary conditions which is relevant to our ensuing discussions. For the simple-quartic lattice with free boundary conditions the Laplacian assumes the form Q = 2(zh + zv)IM ®Itf-
zhHM ® J/v - zvIM ® HJV 104
(22)
Dimers and Spanning Trees: Some Recent Results
1957
where IN is an N x N identity matrix, and HN is the N x N tri-diagonal matrix
000\ 000 000
/1100 1010 0101 HN =
(23)
101 011/
0000 \0000 The eigenvalues of HN are 5 An = 2 c o s mr —,
(24)
n — 0 , 1 , . . . , N — 1.
Then, by diagonalizing Q in the two subspaces of dimensions M and N separately, one obtains the its eigenvalues rnv 1 — cosrmr + 2zv 1 - c o s ^ ~M m = 0,l,...,Al-l, n= 0,l,...,N-l 2zh
(25)
Using (21), one then obtains 4„sin ^ + 4 , „ s i n
^
771=0 71=0
{m,n)j= (0,0)
(26)
and the per-site free energy fsPT(zh,zv)
=
lim M,M^-oo MAf
\
^
IXITM,M(£;
Zh, zv)
I d9 I d(j)\n 4 Zh sin
Jo
Jo
0 + 4 zv sin2
(27)
This leads to /SPT(1,1) =
-G.
(28)
7T
5. Simple-quartic net with a vacancy The similarity between (14) and (27) is striking, since it suggests a connection between the dimer and spanning tree problems. Indeed, Temperley3 has found a bijection between dimer and spanning tree configurations on two related lattices. The bijection has recently been extended to general planar graphs with weighted and/or oriented edges. 16 Here, we describe a version of the bijection relevant to out discussions. 8 105
1958
F. Y. Wu
5.1. Temperley
Injection
Starting from an M x AT simple-quartic net £ with free boundaries, one constructs a dimer lattice CD by i) adding a new site at the midpoint of each edge of £, ii) inserting in each internal face of £ a new site connected to the midpoints of the 4 edges of £ surrounding it, and iii) removing one of the original boundary sites of £ together with its edges incident from the neighboring midpoints inserted in i). Thus, CD has a total of (2M — 1)(27V — 1) — 1 sites consisting of the original MM— 1 sites of £ and the remaining (2M — l){2Af— 1) — MAf new sites. Examples of constructing CD for M. = Af = 3 are shown in Fig. 2. Then we have the following Temperley bijection as contained in results reported in Ref. 16.
O
O
O
<>—1—•—I
O
T—l—°—l—T
it
o
o
It
O
n
it
o
•—'—ii—'—i
•—I—•—I—It
(b)
(c)
(a)
O
it—
o
Fig. 2. (a) A spanning tree lattice C . (b) A dimer lattice Co constructed from C with one corner site removed, (c) A dimer lattice CD constructed from C with one boundary site removed. Open circles denote removed sites.
Temperley bijection: There exists a one-one correspondence between spanning tree configurations on C and dimer configurations on CDTo see that the bijection holds, one observes that to each spanning tree configuration on £, one can construct a unique dimer configuration on CD by first laying a dimer along each tree edge, starting from the edge(s) covering the site of CD which has (have) been removed, and proceed along the spanning tree edges in an obvious fashion. After laying dimers along all tree edges, the remaining sites of CD can then be covered by dimers in a unique way.3 Conversely, starting from each dimer configuration on CD, one constructs a unique tree configuration on £ by drawing bonds (tree edges) along dimers originating from all odd sites. These bonds cannot form close circuits, since otherwise they would have enclosed an odd number of sites of CD which is not permitted in close-packed dimer configurations. This process leads to a unique tree configuration on £. This completes the proof. Examples of the Temperley bijection are shown in Fig. 3. 5.2. Dimer enumeration
on CD
To enumerate dimers on CD which is a simple-quartic lattice with a defect on its boundary, one must start from M, Af odd so that the net of (2.A4 — 1)(2A/" — 1) — 1 sites admits dimer coverings. Let CD and C'D be two dimer lattices derived from 106
Dimers
and Spanning
Trees: Some Recent Results
1959
l m ^ — O J—L^-f]—! = ] — - _ 1—Ft-f
i
[ = i — f =— i L J—L
(a)
(b)
(c)
Fig. 3. Bijections between a spanning tree configuration on C shown in (a) and dimer configurations on two £rj lattices shown in (b) and (c).
C as described in the above. Let the dimer and spanning tree edge weights be the same z/, and zv. Then, the Temperley bijection implies a one-one correspondence between dimer configurations on Co and C'D via the mutual equivalence to spanning trees. A moment's reflection now shows that the two dimer configurations also have identical weights.7 It follows that we have the identity (29)
Z(CD; Zh, zv) = Z(C'D; Zh, zv).
Namely, the dimer generating function is independent of the location of the boundary vacancy. This is a somewhat unexpected result which is difficult to see without the use of the Temperley bijection. For M.,M odd, the aforementioned scheme of orienting lattice edges for the realization of the Kasteleyn criterion no longer holds, since in constructing C\ one needs either M. or H be even. However, using the Temperley bijection it can be shown8 that the dimer generating function for CD is given by Z(CD;zh,zv)
=
tfW-DtflM-Vj.
U V
« *v
). Zh)
(30)
Particularly, for z^ = zv = 1, one has iV dimer (£ D ) = Z(£D; 1,1) = 1
snn
MH
NSPT(C)
M-lAf-l
m=Q n=0
4 sin'* TT-TT + 4 sin"1 ——, ,
2M
27V
( m , n ) ^ (0,0), (31)
where the last line is obtained from (26) with Zh = zv = 1. However, one must note that LB is a (2M — 1) x (27V — 1) lattice with a boundary defect. 6. Finite-size analyses Finite-size expansions of physical quantities associated with two-dimensional lattice models have been of interest both in physics 4 ' 17 and in mathematics. 18 Let ZM N 107
1960
F. Y. Wu
denote the partition function of a lattice model on an M x N lattice. For large M, N one has the expansion In ZM,N
= MNfbulk + JVci + Mc2 + c 3 + o(l),
(32)
where /bulk is the bulk free energy as computed in (14) and (27), ci, c2 are constants independent of M and N, and C3 is a constant which can depend on M and N. In conformal field theory one further computes the limits
S JSJ. ^
- *- + 5 + £ + <"-•>•
(34
>
using which the central charge c can be computed from the values of Ai and A2. These expansions hold for general M and N regardless whether they are even or odd. We have carried out finite-size analyses of our solutions for dimer4 and spanning tree 8 solutions. Here we give the results on the number of dimer configurations (zh = zv = 1) (see also Refs. 11,19,20 for equivalent results). For an M x N dimer lattice with MN = even we use (13) for the number of dimer configurations JVdimer = ZM,AT(£; 1, 1) and obtain
f
~°
7T , /bulk — —
C\ = C 2
- - i l n O L + v/2)
c3 = ^ + I In 2 - ln(l + y/2) + ^
+ £
In ( l + e-(*»-W"j
. (35)
Despite its apparent form, the expression of C3 in (35) is actually symmetric in M and N. The term TTM/24N in C3 now yields the central charge c = 1 upon taken M = N. This agrees with the accepted value for dimer and Ising systems. For an M x TV dimer lattice with one vacant boundary site and both M,N = odd, we use (31) for iVdimer(£i>) and the conversion of M = 2M — 1,N = 27V - 1 to get the desired result. After some algebra, one obtains the same /bulk? ci ?^2 &s in (35), and a new cs given by
n=l
(36) The expression of c 3 , which is again symmetric in A4,N', leads to a new central charge c = —2. Furthermore, the term —^\nN in c'3, which is absent in C3, gives a V^V correction to the dimer enumeration. A concrete example exhibiting this correction has been given in Ref. 8. We remark that Kenyon 18 has found the correction factor to be N3/4 if the vacancy occurs in the interior of the lattice. 108
Dimers and Spanning Trees: Some Recent Results
1961
Acknowledgements It is my pleasure to t h a n k W . T . Lu, R. Shrock, and W.-J. T z e n g for t h e fruitful collaborations that have led to results reported in this paper. T h i s w o r k is s u p p o r t e d in part by National Science G r a n t DMR-9980440.
References 1. H. N. V. Temperley, On the Mutual Cancelation of Cluster Integrals in Mayer's Fugacity Series, Proc. Phys. Soc. 83 3-16 (1964). 2. G. Kirchhoff, Uber die Auflosung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strome gefiihrt wird, Ann. Phys. und Chemie. 72 497-508 (1847). 3. H. N. V Temperley, in Combinatorics: Proceedings of the British Combinatorial Conference (London Math. Soc. Lecture Notes Series # 1 3 , 1974) pp. 202-204. 4. W. T. Lu and F. Y. Wu, Dimer Statistics on the Mobius Strip and the Klein Bottle, Phys. Lett. A259 108-114 (1999). 5. W. J. Tzeng and F. Y. Wu, Spanning trees on hypercubic lattices and nonorientable surfaces, Appl. Math. Lett. 13 19-25 (2000). 6. R. Shrock and F. Y. Wu, Spanning trees on graphs and lattices in d dimensions, J. Phys. A 33 3881-3902 (2000). 7. W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Phys. Lett, to appear and LANL preprint cond-mat/0110035. 8. W. J. Tzeng and F. Y. Wu, Dimers on a simple-quartic net with a vacancy, J. Stat. Phys., to appear. 9. P. W. Kasteleyn, The statistics of dimers on a lattice I. The number of dimer configurations on a quadratic lattice, Physica 27 1209-1225 (1961). 10. H. N. V. Temperley and M. E. Fisher, Dimer problems in statistical mechanics - An exact results, Phil. Mag. 6 1061-1063 (1961). 11. M. E. Fisher, Solution of dimers on a planar lattice, Phys. Rev. 124 1664-1672 (1961). 12. G. Tesler, Matchings in graphs and nonorientable surfaces, J. Comb. Theo. B 78, 198-231 (2000). 13. T. T. Wu, Dimers on rectangular lattices, J. Math. Phys. 3 1265-1266 (1962). 14. F. Harary, Graph Theory (Addison-Wesley 1969). 15. N. L. Biggs, Algebraic Graph Theory (2nd ed., Cambridge Univ. Press, Cambridge, 1993). 16. R. W. Kenyon, J. G. Propp, and D. B. Wilson, Trees and Matchings, LANL preprint math.CO/9903025. 17. H. W. J. Blote, J. L. Cardy, and M. P. Nightingale, Conformal invariance, the central charge, and universal finite-size amplitudes at criticality, Phys. Rev. Lett. 56 742-745 (1986). 18. R. Kenyon, The asymptotic determinant of the discrete Laplacian Acta Math. 185 239-286 (2000). 19. S. D. Ferdinand, Statistical mechanics of dimers on a quadratic lattice, J. Math. Phys. 8 2332-2339 (1967). 20. B. Duplantier and F. David, Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice, J. Stat. Phys. 51 327-434 (1988).
109
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1963-1970 © World Scientific Publishing Company
H Y P E R B O L I C S T R U C T U R E ARISING F R O M A KNOT I N V A R I A N T II: COMPLETENESS
KAZUHIRO HIKAMI Department
of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo 113-0033, Japan.
Received 12 November 2001 We study the geometry of a knot invariant defined in terms of the quantum dilogarithm function. We show t h a t a hyperbolic structure naturally arises in the classical limit of the invariant; t h e completeness conditions can also be identified with the saddle point equations by studying a (l,l)-tangle.
1. Introduction After the discovery of quantum groups, the interest in quantum invariants has been renewed, and we have now infinitely many knot invariants. In contrast to the fact that the Alexander invariant was defined from the homology of the universal Abelian covering, the geometrical meaning of those quantum invariants remains unclear. A key to solving these problems comes from Kashaev's observation 1 ' 2 that the asymptotic behavior of Kashaev's knot invariant, which was later shown3 to coincide with a specific value of the colored Jones polynomial, gives a hyperbolic volume of the knot complement, \\S3\K\\
= ±- lim
^rlog\JN(K)\,
V3 N-*<X> iV
where || • || is the Gromov norm, U3 is the hyperbolic volume of the regular ideal tetrahedron, and JN{K) = VN{K;e2m/N) is given in terms of the colored Jones polynomial V^(if;f) for the knot K (./V-dimensional representation of s^)This paper is a continuation of Ref. 4. Therein we showed that a hyperbolic structure naturally arises from a knot invariant, which was defined by use of the infinite-dimensional representation of the quantum dilogarithm function. In this sense this invariant can be viewed as a non-compact analogue of Kashaev's invariant. We found that the saddle point equations which denote a critical point of our invariant, coincide with the hyperbolicity consistency conditions in gluing ideal polyhedra. One purpose of this paper is to prove that the completeness condition is also given through a classical limit of our knot invariant Ti{K) together with a suitable (l,l)-tangle. See Ref. 5 for a geometrical study of Kashaev's original invariant. Ill
1964
K. Hikami
This paper is organized as follows. We briefly review the construction of our knot invariant in Sec. 2. The essential tool is the S'-operator which solves the five-term relation. In Sec. 3 we give a 3-dimensional picture 4 of our invariant by assigning an oriented tetrahedron to the 5-operator. In the classical limit the tetrahedron can be regarded as an ideal hyperbolic tetrahedron. We clarify how completeness is given from our invariant by studying a related (l,l)-tangle. In Sec. 4 we define a quantum invariant of 3-manifold M based on the ideal triangulation of M. The last section is devoted to concluding remarks. We discuss the relationship between the classical limit of the quantum invariant and the hyperbolic volume. Throughout this paper we use Euler's dilogarithm Li2(z), Rogers' dilogarithm L{z), and the Bloch-Wigner function D(z), which are respectively defined by Li 2 (z) = - f d s l 0 g ( 1 ~ s ) , L{z) = Li 2 (z) + | log(z) log(l - z), Jo s i D(z) = ImLi2(2:) + arg(l - z) log \z\. 2. Knot Invariant We introduce the 5-operator, acting on two spaces V ® V , 5i,2=e*«lfc*7(pi+g2-p2).
(1)
Here we have used the canonical operators, \pj,qk] — —2rySjtk, and
<S>»=exp(7
../T'
, x—),
(2)
\J&+iO 4 sinh(7x) sinh(7ra;) x J which can be regarded as a modular double of the (/-exponential function.6 The ^-operator satisfies the five-term relation, 7-9 S2,3Sl,2 = 5'I I 2 S'I I 35 2 ,3-
(3)
By recursive use of this five-term relation, we see that the i?-operator #12,34 = ( ^ 4 ) - 1 5 1 , 3 ^ 4 ( 5 ^ 3 ) - 1 = Pi,3*2,4/Jl2,34,
(4)
where t a is the transposition on the a-th space and Pa,b is the permutation operator, solves the Yang-Baxter equation (braid relation), #12,34-R34,56#12,34 = #34,56^12,34-^34,56;
^~^---^^-N
=
r^"^--^---^
^
We define a knot invariant from this i?-matrix. For this purpose we give the matrix elements and their classical limit (7 -> 0) in the case of V being the momentum space, p\p) = p\p) with p € l ; = S(jn +P2-P'l)^(p'2-P2
+™ + i 7 ) e 5 M - ^ - ¥ W ^ -
<*(Pi+P2-Pi)exp(- — V ( p ' 2 - p 2 , p i ) \ , 112
P 2
)) (6)
Hyperbolic Structure Arising from a Knot Invariant II 1965 .2i,2
(PUP2\S^\PIP2) = *(Pi -Pi -P2)^l
r~.
- e ^ ~ ^ 17r
* 7 (P2 - P2 -
? - ^ - ^
--17)
~ *(pi - Pi " P2) exp ( 2 T ^ ( P 2 - P^P'I))
Here we have set
+
•
(7)
7T2
F(x, 2 /) = y - L i 2 ( e a : ) - x 2 / ,
(8)
which satisfies
ImF(x, y) = D(l - e x ) + log \ex\ • Im ( ^ ^ )
+ log |e»| • Im f ^ , J / )
Recalling the fact that the volume of an ideal tetrahedron in 3-dimensional hyperbolic space H 3 is written in terms of the Bloch-Wigner function, 10 ' 11 and that the Rogers dilogarithm function is a natural complexification of the hyperbolic volume, 12 ' 13 we can expect that the S'-operator at the critical point is closely connected with an ideal tetrahedron in H 3 . We can define the knot invariant from the enhanced Yang-Baxter operator 14 {R,H,a,p), n(L) = a-w^(3-n
Tr 2 ,..., n (b A (0(l ® / x ^ " " 1 ) ) ) .
(10)
Here £ denotes the braid group representation with n strands of link L, and 6^(£) is to substitute the operator R (4) as a braid generator. We have used w(£,) as writhe, and "- + -> a
j" 2 +-T 2
_2L±J.fit
_
7e
I
2
(1 - e 'T)(l - e2l7r /T)
^
Note that - ri(if) is an invariant for (l,l)-tangles, and that we do not take trace over the first space. 3. Hyperbolicity and Completeness We have clarified in Ref. 4 how the hyperbolic structure naturally arises from the knot invariant T\(K) in the classical limit 7—^0. Therein, motivated by the fact that the S'-operator solves the five-term relation (3), the S-operator is identified in the classical limit with an ideal tetrahedron in H 3 ,
=
/
M " /
r i
(Pi.PalS-Vi.Pa)
113
1966
K. Hikami
Here momenta pa and p'a are assigned to every face. Both oriented ideal tetrahedra have modulus z = ep^~P2, and every dihedral angle is fixed as a function of the modulus, z\{z) = z, 22(2) = 1 — z - 1 , a n d zz{z) = (1 — z)_1> opposite edges having the same angles. The integral with respect to momentum pa is interpreted as gluing two faces having the same momentum to match the orientation of every edge. One sees that the five-term relation (3) is simply realized as the 2 •<->• 3 Pachner move. Correspondingly, the il-matrix (4) can be depicted as an oriented ideal octahedron,
<0\Rtf) =
(12)
Here we have pi + p3 = p\ + p2 and p'2 +p'^ = p'3 + p^. The picture on the right denotes the projection of the octahedron viewed from the top of the octahedron, and a,i is the dihedral angle around a central axis "," ai = z^{ez~Pl), a2 = z2(eP3~z), a>3 = zz{ew~Pi), and 0,4 = z2{eP2~w), satisfying the consistency condition, I — 11
'
'1; ',
I _ gPl-p'i+P4-P4
ePl-p'i+P4-P4
e~ = e-P4 — ePl-p'1+Pi-P4-P2 ~
'
^e
= e~p'3 _
gPl-Pi+Pi-P't
-Pi
(13)
Note that we have symmetry of the i?-matrix, and that the inverse braiding generator R"1 is also given by an oriented ideal octahedron, (Pl,P2,P3,P4\R\p'l,P2,P3>P4) {Pl,P2,P3,P4\R~1\p'l,P2,P3,P4)
=
{P4,P3,P2,Pl\R\P4,P3,P2,Pl},
= P4,]'3).
Our main claim in Ref. 4 is that the saddle point conditions which are derived from the classical limit of T\{K), exactly coincide with the hyperbolicity consistency conditions in gluing ideal tetrahedra. Generally, to endow the hyperbolic structure in a 3-manifold M which is constructed by gluing together a finite collection of ideal tetrahedra, we should check the completeness condition besides the hyperbolicity consistency conditions. 10 We shall show that this completeness condition is fulfilled by considering the knot invariant as coming from a constituent (l,l)-tangle. In computing T\{L) from a (l,l)-tangle, we cut the link L at a point which is located on an alternating segment of L (left figure below), Pi '31s-
P5„
\
P3 P4
/
/ / i/
\
P2 114
Hyperbolic Structure Arising from a Knot Invariant II
1967
We glue together faces, ps and p 4 , and part of the developing map is written as the right figure above (see Ref. 5 for some discussion). The completeness condition can be read from this picture as T^-
= T-~
•
(14)
On the other hand, the contribution from this segment to the invariant is given by /
dxdu{p3,x\S
1
1
\y,Pi){p5,v\S
\u,p3) {z,p2\S\p4,x)
(u,p4\S\p6,w)\p3=pi=p.
Here we have assumedp 3 = PA = P as we are studying the (l,l)-tangle. Substituting Eqs. (6)-(7) as the limit 7 —» 0, we obtain the saddle point condition as 1 _
Ppi
T ^ ^
-x
= !•
( 15 )
When we set p3 = p^ = P = ±00, we find that the saddle point equation (15) coincides with the completeness condition (14). We can see that other completeness conditions can be deduced from Eq. (14) with the help of the hyperbolic consistency conditions (13) in constructing the i?-matrix. As a result, by constructing the invariant of link L from a constituent (l,l)-tangle and substituting a specific value therein, we can see a correspondence between the completeness conditions and saddle point equations. Combining our previous result4 that the hyperbolicity consistency conditions coincide with the saddle point equations, we can conclude that the invariant T\{L) with the 3-dimensional picture (11) gives an ideal triangulation of the knot complement, and that Eq.(9) will indicate a coincidence between the asymptotic value of our invariant at the critical point and the hyperbolic volume of the knot complement. 4. Q u a n t u m Invariant of Manifold We define a quantum invariant of manifold M in terms of the 5-operator (1). When 3-manifold M admits a hyperbolic structure of finite volume, the 3-manifold M is constructed by gluing a finite collection of oriented ideal tetrahedra. We assign an S'-operator to each oriented ideal tetrahedron, and we define the partition function of M by 1 5 Z(M) = jj&p n ^ . P i l ^ l P f c . P i ) .
(16)
Here we need to assume a constraint for p which corresponds to the completeness condition of M. As was studied in the previous section for the case of the knot invariant Ti(L), the hyperbolicity consistency conditions are given as the saddle point equations in the integral over p, while the completeness conditions are controlled by considering a (l,l)-tangle of link L. It should be noted that the idea of assigning a solution of the five-term relation to a tetrahedron, 1 6 ' 1 7 where the Regge action is derived from the asymptotic value 115
1968
K. Hikami
of the classical 6j-symbol for large j , is well known. We have rather clarified that the S-operator (1) denotes a quantization of the hyperbolic ideal tetrahedron. 4.1.
Examples
We consider knots K = 4i and 5 2 ,
o As is well-known, these knots are hyperbolic, and the complements of the knots can be constructed by gluing ideal tetrahedra (see, e.g. Refs. 18,19). Indeed the ideal triangulation can be done following Ref. 4, and we get the quantum invariant by assigning S-operators to each oriented ideal tetrahedron, Z ( S 3 \ 4 i ) = /dp5p 4 _p 2=Pl _p3 (J>\,P2\S\PZ,PA) Z(S3\52)
=
dp6P5-p4=p2-P6
{p4,P3|£'~ 1 |p2,Pi),
(pi,p 5 |S'~ 1 |P4,P3)
X (P2,P4|5'"1|P6,P5) (P3,P6\S~1\pi,P2)The constraint 6... represents the completeness condition. In the small-7 limit, these integrals respectively reduce to Z(S3 \ 4 0 - / d z e x p - i - (Li2(e-X) - Li 2 (e*)), J 217 Z(S3 \ 52) ~ JJdxdyexp
^
( y - Li 2 (e*-*) - 2 Li 2 ( e -») - y(y -
x)).
After applying the saddle point method whose conditions exactly coincide with the hyperbolicity gluing conditions and completeness conditions, we see 3
7 og {
\
("2.029883212819307, for K = 4X ) ~ j 2.828122088330783 + 3.024128376509301 i, for K = 5 2
Comparing with a table in Ref. 20 (it is necessary to multiply the Chern-Simons terms there by 27r2), this result suggests the "VCS conjecture", 3 ' 5 ' 1 5 ' 2 1 2 7 log Z(M) ~ Vol(M) + i CS(M) = VCS(M),
(17)
where Vol and CS respectively denote the hyperbolic volume and the Chern-Simons invariant of M. 5. Concluding Remarks We have studied how the knot invariant T\(L) is related with the hyperbolic geometry H 3 in the limit 7 — ^ 0 . We have shown that not only the hyperbolicity consistency conditions but also the completeness conditions can be derived from 116
Hyperbolic Structure
Arising from a Knot Invariant
II
1969
t h e saddle point equations when we consider t h e invariant of a link L as resulting from the invariant of a ( l , l ) - t a n g l e . Based on the 5-operator (1), we h a v e defined the p a r t i t i o n function (16) of 3-manifold M. Once M is t r i a n g u l a t e d into a finite collection of ideal tetrahedra, t h e volume Vol(M) is given by a s u m m a t i o n , £ \ D(zi), where modulus Zi satisfies a set of hyperbolicity and complete conditions. Recalling t h a t t h e 5-operator and its imaginary part respectively reduce t o t h e Rogers dilogarithm function L(z) and t h e Bloch-Wigner function D(z) a t t h e critical point (9), t h e conjecture (17) seems t o be reasonable. However, the Rogers dilogarithm function L(z) is a multi-valued function of z, with singularities a t 0 a n d 1, a n d t h e value on t h e universal abelian cover of C \ { 0 , 1 } is given with a n integer pair ( c i , c 2 ) a s 1 3 L(z) + ^ ( c i log(l — z) + C2 log(z)). In computations such as for K = 6 i , we need such t e r m s t o get "correct" answers. It remains for future studies t o discover how to specify t h e branch of L(z) in the small-7 limit. Prom a physical point of view, t h e hyperbolic geometry or t h e Euclidean AdS receives much attention based on t h e A d S / C F T correspondence. As it is well known t h a t the Einstein-Hilbert action can b e r e w r i t t e n in t e r m s of t h e CS a c t i o n , 2 2 ' 2 3 our result which relates the partition function Z(M) with H 3 a n d CS will be promising in further studies of q u a n t u m gravity. Acknowledgement This work is supported in p a r t by t h e S u m i t o m o Foundation. T h e author would like to thank Hitoshi Murakami for useful communications. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
R. M. Kashaev, Mod. Phys. Lett. A 1 0 , 1409 (1995). R. M. Kashaev, Lett. Math. Phys. 39, 269 (1997). H. Murakami and J. Murakami, Acta Math. 186, 85 (2001). K. Hikami, Int. J. Mod. Phys. A 1 6 , 3309 (2001). Y. Yokota, preprint (math.QA/0009165) L. D. Faddeev, Lett. Math. Phys. 34, 249 (1995). L. D. Faddeev, R. M. Kashaev, and A. Yu. Volkov, Commun. Math. Phys. 219, 199 (2001). L. Chekhov and V. V. Fock, Theor. Math. Phys. 120, 1245 (1999). S. L. Woronowicz, Rev. Math. Phys. 12, 873 (2000). W. P. Thurston, The Geometry and Topology of Three-Manifolds (Princeton, 1980). J. Milnor, Bull. Amer. Math. Soc. 6, 9 (1982). W. Z. Neumann and D. Zagier, Topology 24, 307 (1985). W. D. Neumann, in Topology '90 (de Gruyter, Berlin, 1992), pp. 243-271. V. Turaev, Invent. Math. 92, 527 (1988). K. Hikami, Nucl. Phys. B616, 537 (2001). G. Ponzano and T. Regge, in Spectroscopic and Group Theoretical Methods in Physics (North-Holland, 1968), pp. 1-58. V. G. Turaev and O. Y. Viro, Topology 3 1 , 865 (1992). M. Takahashi, Tsukuba J. Math. 9, 41 (1985). 117
1970 K. Hikami 19. 20. 21. 22. 23.
J. Weeks, SnapPea, h t t p : / / t h a m e s . n o r t h n e t . o r g / w e e k s / . P. J. Callahan, J. C. Dean, and J. R. Weeks, J. Knot Theory Ramif. 8, 279 (1999). S. Baseilhac and R. Benedetti, preprint (math.GT/0101234). A. Achucarro and P. K. Townsend, Phys. Lett. B180, 89 (1986). E. Witten, Nucl. Phys. B 3 1 1 , 46 (1988/1989).
118
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1971-1977 © World Scientific Publishing Company
EXOTIC GALILEAN S Y M M E T R Y A N D T H E HALL E F F E C T
C. DUVAL Luminy, Case 907, F-13 288 MARSEILLE
(France)
P. A. HORVATHY Laboratoire de Mathematiques et de Physique Thiorique, Universite Pare de Grandmont, F-37 200 TOURS (France)
de Tours
Received 5 November 2001 The "Laughlin" picture of the Fractional Quantum Hall effect can be derived using the "exotic" model based on the two-fold centrally-extended planar Galilei group. W h e n coupled to a planar magnetic field of critical strength determined by t h e extension parameters, the system becomes singular, and "Faddeev-Jackiw" reduction yields the "Chern-Simons" mechanics of Dunne, Jackiw, and Trugenberger. The reduced system moves according to the Hall law.
1. Introduction In his seminal paper1 Laughlin argued that the Fractional Quantum Hall Effect2 could be explained as condensation into a collective ground state, represented by the lowest-Landau-level wave functions /(z)e-W 2 / 4 ,
(1)
where the complex N-vector z denotes the positions of N polarized electrons in the plane; f(z) is analytic. The fundamental operators are zf — zf, and zf = 2dzf satisfy \z, z\ = 2. The quantum Hamiltonian only involves the potential V(z, z) suitably quantized with the choice of an ordering for the non-commuting operators z and z. Our results 3 presented here say that the Laughlin picture can actually be obtained from first principles, namely using the two-fold central extension of the planar Galilei group. This latter has been known for some time, 4 , 5 but has long remained a kind of curiosity, since it had no obvious physical use: for a free particle of mass m, the extra structure related to the new invariant k leaves the usual motions unchanged, and only contributes to the conserved quantities. 3 ' 5 ' 6 Let us *TaIk given by P. A. Horvathy at the Joint APCTP- Nankai Symposium. Tianjin (China), Oct.2001 119
1972 C. Duval & P. A. Horvdthy mention that our "exotic" theory is in fact equivalent to Quantum Mechanics in the non-commutative plane, 7 with non-commutative parameter 9 = k/m2. Coupling an "exotic" particle to an electromagnetic field, the two extension parameters, k and m, combine with the magnetic field, B, into an effective mass, m*, given by (4); when this latter vanishes, the consistency of the equations of motion requires that the particle obey the Hall law. Interestingly, for m* = 0, Hamiltonian reduction 8 yields the "Chern-Simons mechanics" considered before by Dunne, Jackiw and Trugenberger. 9 The reduced theory admits the infinite symmetry of area-preserving diffeomorphisms, found before for the edge currents of the Quantum Hall states. 10
2. Exotic particle in a gauge field Let us consider the action
/
(p — A) -dx — hdt + -px
dp,
(2)
where (V, A) is an electro-magnetic potential, the Hamiltonian being h = p2/2m + V. The term proportional to the non-commutative parameter 9 is actually equivalent to the acceleration-dependent Lagrangian of Lukierski et al. 6 The associated EulerLagrange equations read m*Xi = Pi — mOsijEj, (3) Pi — *-*% i t5 Sij Xj, where we have introduced the effective mass
m*
=m(l-0B).
(4)
The velocity and momentum are different if 9 ^ 0. The equations of motions (3) can also be written as / 0 dh W«(3^
=
where
Ko)
9
1
0\
-9 0
0 1
-10
0
0
5
(5)
-1-B0
Note that the electric and magnetic fields are otherwise arbitrary solutions of the homogeneous Maxwell equation dtB + SijdiEj = 0, which guarantees that the twoform u = hu>a/3d^a A d£p is closed, duj = 0. 120
Exotic Galilean Symmetry and the Hall Effect 1973 W h e n m* ^ 0, t h e determinant det (wQ/g) = (1 - 6B) = (m*/m) is nonzero a n d t h e m a t r i x (uiap) m (5) can be inverted. Then the equations of motion (5) (or (3)) t a k e the form £ a = {£ Q , h}, with the standard Hamiltonian, but with the new Poisson bracket {/, g} = {ui~1)ai3dafdpg which reads, explicitly,
{/,
m m*
+6
dx
dp
Ox
dp
^L_d9__^9__d£ Kdx\
dx2
dxi
+B
8x2
df_dg_
dg_df_
dpi dp2
dpi dp2
(6)
Further insight can b e gained when the magnetic field B is a (positive) nonzero constant, which t u r n s out t h e most interesting case, and will b e henceforth assumed. ( T h e electric field Ei = — diV is still arbitrary). Introducing t h e new coordinates ^4% — %i \ B mr
l - W —
Eij Pj,
(7)
1 V% —
-^BeijQj,
will allow us to generalize our results in Ref. 3 from a constant to any electric field. Firstly, the C a r t a n one-form 1 2 in the action (2) reads simply PidQi — hdt, so t h a t t h e symplectic form o n phase space retains the canonical guise, w — dPiAdQi. T h e price to pay is t h a t the Hamiltonian becomes rather complicated. 3 T h e equations of motion (3) are conveniently presented in terms of the new variables Q a n d t h e old m o m e n t a p, as Ei l°ci — £ii n T \
Pi
enB- m
6ij
El
E± B
£jk
m
rrr
B (8)
N o t e t h a t all these expressions diverge when m* tends to zero. W h e n the magnetic field takes the particular value B = Br =
1
d'
(9)
t h e effective mass (4) vanishes, m* = 0, so that det(w a ^) = 0, and the system becomes singular. T h e n the t i m e derivatives £ a can no longer be expressed from t h e variational equations (5), a n d we have resort to "Faddeev-Jackiw" reduction. 8 I n accordance with t h e D a r b o u x theorem (see, e.g., Ref. 12), the C a r t a n one-form in (2) can be written, u p to a n exact term, as i? — hdt,
with
1 1 •& = (pi — -Bc Eij Xj)dxi + -9eij p%dpj = PidQi,
(10)
where t h e new coordinates read, consistently with (7), 1 Qi
Xi
B, 121
•
£ijPj,
(11)
1974
C. Duval & P. A.
Horvdthy
while the Pi = — ^BcSijQj are in fact the rotated coordinates Qi. Eliminating the original coordinates x and p using (11), we see that the Cartan one-form reads PidQi - H(Q,p)dt, where H(Q,p) = p2/(2m) + V(Q,p). As the pi appear here with no derivatives, they can be eliminated using their equation of motion dH(Q,p)/dp = 0, i. e., the constraint Pi
Eij-E'j 0. (12) m Bc A short calculation shows that the reduced Hamiltonian is just the original potential, viewed as a function of the "twisted" coordinates Q, viz.
H = V(Q).
(13)
This rule is referred to as the "Peierls substitution". 3 ' 9 Since d2H/dpidpj = 5ij/m is already non singular, the reduction stops, and we end up with the reduced Lagrangian Lied = ^QxQ~-V(Q),
(14)
supplemented with the Hall constraint (12). The 4-dimensional phase space is hence reduced to 2 dimensions, with Q\ and Q2 in (H) as canonical coordinates, and reduced symplectic two-form ujred = ^Bc£ijdQi A dQj so that the reduced Poisson bracket is dG flFN dF fFC\ = _ J1- /dF ( i L idG £_0G { B \dQ dQ ~ dQi dQ c x 2 2l^'Wred BAOOA dO.,. 3Qi dQJ' ' The twisted coordinates are therefore again non-commuting, { Q i , Q 2 }I rred e d = - 0 = - ^ Rc
(16)
The equations of motion associated with (14), and also consistent with the Hamilton equations Qi = {Qi, H}Ted, are given by Qi=eiM,
(17)
Bc
in accordance with the Hall law (compare (8) with the divergent terms removed). Putting Bc — 1/6, the Lagrangian (14) becomes formally identical to the one Dunne et al. 9 derived letting the real mass go to zero. Note, however, that while Q denotes real position in Ref. 9, our Q here is the "twisted" expression (11), with the magnetic field frozen at the critical value Bc = 1/0. 3. Infinite symmetry It has been argued 11 that the physical process which yields the Fractional Quantum Hall Effect actually takes place at the boundary of the droplet of the "Hall" liquid: owing to incompressibility, the bulk can not support any density waves, but there are chiral currents at the edge. These latter fall into irreducible representations 122
Exotic Galilean Symmetry and the Hall Effect 1975
of the infinite dimensional algebra Wi+oo,10 which is the quantum deformation of Woo, the algebra of classical observables which generate the group of area-preserving diffeomorphisms of the plane. Our reduced model is readily seen to admit iUoo> the classical counterpart of W\+00, as symmetry. To see this, let us remember that, as argued by Souriau,12 and later by Crnkovic and Witten, 1 3 it is convenient to consider the space of solutions of the equations of motion (Souriau's "espace des mouvement^ [= space of motions]), denoted by M. For a classical mechanical system, this is an abstract substitute for the classical phase space, whose points are the motion curves of the system. The classical dynamics is encoded into the symplectic form fi of M.. It is then obvious that any function /(£) on Ai is a constant of the motion. (When expressed using the positions, time, and momenta, such a function can look rather complicated). Any such function / (£) generates a Hamiltonian vectorfield Z^ on M through the relation -dpf
= %VZV.
(18)
The vector field Z^ generates, at least locally, a 1-parameter group of diffeomorphisms of M.. All diffeomorphisms of Ad which leave the symplectic form ft invariant form an infinite dimensional group, namely the group of symplectomorphisms of M. Any symplectic transformation is a symmetry of the system : it merely permutes the motions curves. For the reduced system above, the reduced phase space is two dimensional. The space of motions is therefore locally a plane. (Its global structure plainly depends on the details of the dynamics). Now, for any orientable two dimensional manifold, the symplectic form is the area element; it follows that the reduced system admits the group of area-preserving transformations as symmetry. 4. Quantization Let us conclude our general theory by quantizing the coupled system. Again, owing to the exotic term, the position representation does not exist. Introducing the complex coordinates
' = ^{Qi
(19)
x
IfB
-*Qa) + -j={ -iPi
- P2)
the two-form dPiAdQi on 4-dimensional unreduced phase space becomes the canonical Kahler two-form of C 2 , viz OJ = (2i)~1(^dz A dz + dw A dw). choosing the antiholomorphic polarization, the "unreduced" quantum Hilbert space, consisting of the "Bargmann-Fock" wave functions ip(z, z, w, w) = f(z, w)e-i{z*+w™\ 123
(20)
1976
C. Duval & P. A. Horvdthy
where / is holomorphic in both of its variables. The fundamental quantum operators, zf = zf, Zf = 2dzf, wf = wf,wf
(21)
= 2dwf,
satisfy the commutation relations [z,z\ = [w,w\ = 2, and \z,w\ = [l,w] = 0. We recognize here the familiar creation and annihilation operators, namely a*. = z, a*„ = w, and az = dz, aw = dw. Using (7), the (complex) momentum p — p\+ip2 and the kinetic part, ho, of the Hamiltonian become, respectively, • rnB _ p=—i\l w V m*
and
B ho = - — w w . 2m*
(22)
ForTO*7^ 0 the wave function satisfies the Schrodinger equation idtf = hf, with h = ho + V. The quadratic kinetic term here is ho =
(wul+ww) = -—(ww + l). (23) K 4m* v ' 2m*y ' ' The case when the effective mass tends to zero is conveniently studied in this framework. On the one hand, in the limit m* -v 0, one has z -> VBQ, W -> 0, (24) where Q = Q\ +1Q2, cf. (7); the 4-dimensional phase space reduces to the complex plane. On the other hand, from (22) and (21) we deduce that 777*
$ = w = 2dw. (25) mB The limitTO*—> 0 is hence enforced, at the quantum level, by requiring that the wave functions be independent of the coordinate w, i.e., dwf = 0,
(26)
yielding the reduced wave functions of the form *(z,z) = f(z)e-i'*,
(27)
where / is a holomorphic function of the reduced phase space parametrized by z. When viewed in the "big" Hilbert space (see (20)), these wave functions belong, by (23), to the lowest Landau level. 2 ' 3 ' 14 Using the fundamental operators z a n ? given in (21), we easily see that the (complex) "physical" position x = x\ + 1x2 and its quantum counterpart x, namely 1
/
x=-=[z+J—w), WBn V
/ TO \
V m*
I
^
1
/
/ m „„ \
x=-m=(z+J—2dw), JBa
V
V m*
I
.„„,
28
manifestly diverge whenTO*—> 0. Positing from the outset the conditions (26) the divergence is suppressed, however, leaving us with the reduced position operators xf = Qf = -^=zf,
*f=Qf 124
= -?=dzf,
(29)
Exotic Galilean Symmetry and the Hall Effect
1977
whose c o m m u t a t o r is [Q,Q] = 2/Bc, cf. (16). In conclusion, we recover the "Laughlin" description (1) of t h e ground states of the FQHE in Ref. 2. Quantization of the r e d u c e d H a m i l t o n i a n (which is, indeed, the potential V(z,z)), can be achieved using, for instance, anti-normal o r d e r i n g . 9 ' 1 4
Acknowledgements We are i n d e b t e d t o Professors J. G a m b o a , Z. Horvath, R. Jackiw, L. M a r t i n a a n d F . Schaposnik for discussions. P. H. would like to thank Prof. Mo-lin Ge for hospitality e x t e n d e d t o h i m a t Nankai I n s t i t u t e of Tianjin (China).
References 1. R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 2. See, e.g., Quantum Hall Effect, Ed. M. Stone, World Scientific Singapore (1992); S. M. Girvin, The Quantum Hall Effect. New excitations and broken symmetries. Les Houches Lectures (1998). 3. C. Duval and P. A. Horvathy, Phys. Lett. B 479, 284 (2000), hep-th/0002233; Exotic galilean symmetry in the non-commutative plane, and the Hall effect, hep-th/0106089, to appear in Journ. Phys. A; C. Duval, Z. Horvath, and P. A. Horvathy, Int. Journ. Mod. Phys. B 15, 3397 (2001), cond-mat/0101449. 4. J.-M. Levy-Leblond, in Group Theory and Applications (Loebl Ed.), II, Acad. Press, New York, p. 222 (1972). 5. A. Ballesteros, N. Gadella, and M. del Olmo, Journ. Math. Phys. 33, 3379 (1992); Y. Brihaye, C. Gonera, S. Giller and P. Kosiriski, hep-th/9503046 (unpublished); D. R. Grigore, Journ. Math. Phys. 37, 240 and 460 (1996). 6. J. Lukierski, P. C. Stichel, W. J. Zakrzewski, Annals of Physics (N. Y.) 260, 224 (1997). 7. J. Gamboa, M. Loewe, and J. C. Rojas, hep-th/0010220 and hep-th/0106125; V. P. Nair and A. P. Polychronakos, Phys. Lett. B505, 267 (2001); Z. Guralnik, R. Jackiw, S. Y. Pi and A. P. Polychronakos, Phys. Lett. B517, 450 (2001). 8. L. Faddeev and R. Jackiw, Phys. Rev. Lett. 60, 1692 (1988). 9. G. Dunne R. Jackiw and C. A. Trugenberger, Phys. Rev. D41, 661 (1990); G. Dunne and R. Jackiw, Nucl. Phys. B (Proc. Suppl.) 33C 114 (1993). 10. S. Iso, D. Karabali, B. Sakita, Phys. Lett. B296, 143 (1992); A. Cappelli, C. A. Trugenberger, G. R. Zemba, Nucl. Phys. B 396, 465 (1993); A. Cappelli, G. V. Dunne, C. A. Trugenberger, G. R. Zemba, Nucl. Phys. B (Proc. Suppl). 33C, 21 (1993); J. Martinez and M. Stone, Int. Journ. Mod. Phys. B26, 4389 (1993). 11. D. H. Lee and X. G. Wen, Phys. Rev. Lett. 66, 1765 (1991); M. Stone, Ann. Phys. (N.Y.) 207, 38 (1991). X. G. Wen, Int. Journ. Mod. Phys. B 6, 1711 (1992). 12. J.-M. Souriau, Structure des systemes dynamiques, Dunod, Paris (1970); Structure of Dynamical Systems, a Symplectic View of Physics, Birkhauser (1997). 13. C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, in 300 years of gravitation, Ed. S. Hawking and W. Israel, Cambridge U.P. (1987). 14. S. Girvin and T. Jach, Phys. Rev. B 2 9 , 5617 (1984); G. Dunne, Ann. Phys. 215, 233 (1992).
125
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1979-1986 © World Scientific Publishing Company
THE THREE-STATE C H I R A L CLOCK MODEL*
BAI-QI JIN Physics Department
and Centre for Nonlinear Studies, Hong Kong Baptist Kowloon Tong, Hong Kong
University
HELEN AU-YANG and J A C Q U E S H.H. P E R K t Physics Department,
Oklahoma State University,
Stillwater,
OK 74078,
U.S.A.
Received 29 November 2001 We give a brief summary of our recent works on the three-state chiral clock model. In these works, we use improved effective field theories with clusters being strips, infinite in t h e chiral direction and finite in the non-chiral direction. Hence, effective-field transfer matrix methods can be employed in these studies. The effective fields are determined by the Gibbs-Bogoliubov free energy variational principle, leading to Weiss or Bethe approximations in different studies respectively. By systematic improvement of these approximations, i.e. widening the strips, these studies point to the conclusion that there is no Lifshitz point existing at finite non-zero chirality.
1. Introduction The three-state chiral clock model was introduced independently by Ostlund 1 and Huse. 2 It is the simplest model with only nearest-neighbor interactions which exhibits spatially modulated phases. These spatially modulated phases occur diversely in physical systems. 3 The reduced Hamiltonian for this model on the two-dimensional square lattice is -/3if({n i , j },A) =
T^
2-7T ,
^ K cos — [riij -riij+i+A) n
.
T^
277 .
+ Kt cos — (jiij
.
-ni+ij) (1)
where (3 = 1/kgT. Prom the symmetry within this model, we can restrict ourselves to 0 < A < 1/2 without losing generality. Ostlund used free-fermion analysis, which is valid for low temperature and A close to 1/2, to show that there are incommensurate phases in this model. This fact makes the model interesting for the study of commensurate-incommensurate phase transitions and hence it has been the focus of considerable theoretical efforts. "This work has been supported in part by NSF G r a n t s No. PHY 97-22159, PHY 97-24788 and PHY 01-00041. t Email address [email protected] 127
1980
B.-Q. Jin, H. Au-Yang
& J. H. H. Perk
The model (1) has been studied by finite-size scaling methods, 4 - 8 Monte-Carlo simulation, 9 hierarchical lattice approximation,10 Monte-Carlo renormalization group 11 and series expansion methods. 12 " 14 Important analytical predictions using domain-wall arguments and general topological ideas also have been presented. 15-17 In spite of all these efforts, several features remain controversial, for example, the existence of a Lifshitz point at A ^ 0 in the phase diagram of this model. Haldane et al., 15 Schulz,16 and Von Gehlen and Rittenberg 7 argue against the idea of a Lifshitz point at A ^ 0, while Howes, 12 Huse and Fisher,17 Selke and Yeomans, 9 Duxbury et al., 4 and Martins and Tsallis 10 are presenting arguments for it. Apart from this controversy over a qualitative feature, there are also uncertainties concerning the nature of various phase transitions in this model. In order to shed more light on these problems, we used improved effective field theories with clusters to be taken as strips which are infinite in the chiral direction and finite in the non-chiral direction. This treatment is equivalent to separating the original two-dimensional square lattice into many identical decoupled strips with effective fields on their boundaries and treating interactions within them exactly. Effective-field transfer matrix methods 18 can be successfully used in such striprelated calculations. Obviously, within an improved effective field theory, we have to pay serious attention to i) how to put the effective fields on the boundary (so as to partially include the effects of the out-of-cluster part of original system) and ii) how to relate the typical order parameters of the finite-strip system to ones of the original system. These two aspects determine whether the approximate critical points obtained will be converging to the true ones and how fast the convergence will be. Currently, the most-commonly used effective field theories employ the Gibbs-Bogoliubov free energy variational principle, resulting in the Weiss and Bethe approximations. It has been found that even an infinite chain with effective fields (which are determined from free energy considerations) on the boundaries can qualitatively improve the simple effective field results. 19 More interestingly, as advocated by Suzuki, it is possible to apply the coherent anomaly method (CAM) 18 to well-chosen sequences of effective field theories. By systematically treating wider and wider strips—i.e. more and more interactions are treated exactly—one obtains better and better approximations to the exact phase diagram of the original physical system and an excellent extrapolation to the exact results can be expected from these successive approximations, if the strips become wide enough. This paper is organized as follows. In Section 2, we present our analysis of effective field theories based on the Gibbs-Bogoliubov free energy variational principle. 20 In Section 3, we first show how the approximate wavevector-dependent susceptibility is obtained in two series of effective field theories with either Weiss or 128
The Three-State Chiral Clock Model
1981
Bethe approximations, resulting in two series of Lifshitz point approximants. Prom these a new series is constructed showing that possibly no Lifshitz point exists at finite (non-zero) chirality.21 A brief summary is given in Section 4. 2. Effective Field Theory from Free Energy Considerations The approximate free energy FMF is obtained by the use of the Gibbs-Bogoliubov inequality F < FMF = mm{F0
+ (H - H0)),
(2)
where F is the exact free energy of t h e original system with H being t h e original Hamiltonian. HQ is a trial Hamiltonian and FQ is the exact free energy of t h e s y s t e m defined by Ho. T h e average (• • •) is carried out in the ensemble defined b y Ho a n d this convention will be used throughout this paper. For b o u n d a r y spins, it is more convenient to introduce the vector notation 2TT
Si,j = ( cos —riij, —n , id
.
2TT
sin— s i n y nitj ) .
(3)
H is given in Eq. (1) and HQ is denned as follows: -0HQ = ^2 Kn cos — ( n ; j - mij+1 + A) ~3 JV,-l(p+l)JV-2 +
Yl
J2
p=0
k=pN
N,-l
h-\
^2KtcosY(nkj-nk+hj) j
+ 2^i y ' ^tVk " (SpJV-l,p'L+fc + SpN,p'L+k),
(4)
p,p'=0 *:=0
where 0 < i < NSN — 1, 0 < j < NSL — 1, periodic boundary conditions are imposed on both directions, and f3 = l/ksT. The trial Hamiltonian HQ consists of Ns independent strips of width N and length NSL with effective boundary fields {Vj = (Vji-irlj2)} having period L to replace the exact interactions between strips. To find a good approximation for the free energy, we use Eq. (2) to find the minimum conditions which {r}j} should satisfy. The necessary minimum conditions can be simplified as Vj = (so,j) = mj.
(5)
The corresponding approximate free energy per site / M F can be given as Kt L~1 fMF = fo + jjj-jJ2(2llj-mi-mJ-nii)> P
(6)
j=0
where /o is the free energy per site of the system defined by HQ and can be calculated by the effective transfer matrix method. 18 ' 20 129
1982 B.-Q. Jin, H. Au-Yang & J. H. H. Perk
It is easy to see that the effective fields in our trial Hamiltonian are essentially the thermal averages of the strip boundary spins whose configuration can be used to characterize the related phase. Eqs. (5) can be solved by iteration methods and interested readers can consult our full paper 20 for technical details. From physical considerations, we may expect three types of solutions to Eqs. (5), i.e. i) the disordered solution with (77 = 0), ii) the ordered solution with (77 ^ 0) which can be obtained by setting L = 1 in Eqs. (5), and iii) modulated solutions with unequal effective fields, i.e. L > 1. The thermodynamically stable phase is the one that gives the absolute minimum free energy for all different solutions with all possible L. Hence, for 0 < A < 1/2, we can expect that the disordered solution gives the lowest approximate free energy for the disordered phase and the ordered solution for commensurate phase. In the modulated phase, one of the modulated solutions should give the lowest approximate free energy and the choice of solution may vary from point to point. The numerical results are summarized as follows. We obtain A L ( 1 ) « 0.3143, A L (2) w 0.2883, A L (3) ra 0.2770, A L (4) « 0.2709 for Kn = Kt and A L (1) » 0.2258, A L (2) < 0.2156 for Kn = WKt, where the notation AL(N) is used to denote the approximate Lifshitz point from the effective field theory for a strip of width N. Hence, we can safely claim that the Lifshitz point Az,(N) located by this approximation is systematically decreased when the width N becomes larger. The result for different ratios of Kn/Kt also coincides with our intuition that the larger Kn/Kt leads to faster convergence. Its possible explanation is discussed in our papers 20 and 21. As reviewed by Wu, 22 simple mean-field theory predicts a first-order phase transition in the three-state Potts model, which is equivalent to the A = 0 three-state chiral clock model. Our effective field theory with finite-width strip also predicts a first-order phase transition for 0 < A < | which is characterized by a sudden change of spin profiles (described by the thermal average of the central-row spins in our effective field theory) due to the discontinuity of the effective fields when the system crosses the critical point. However, this artificial feature of the effective field theory can be overcome by systematically improving the effective field approximation. 20 When N —>• 00, these effective fields {r)j} (which are non-vanishing but have no direct physical meanings in the original problem) should give an innnitesimaUy small effect on the spin profiles (which should approach zero) and on the specific heat. Hence, the extrapolation of these effective field approximations would be able to give the correct nature of the phase transition, i.e. a continuous phase transition.
3. Effective Field Theory from Susceptibility Considerations When the system changes from the disordered phase into the incommensurate phase as the temperature is lowered, the peak of the wavevector-dependent susceptibility 130
The Three-State Chiral Clock Model
1983
also changes into a divergence. Hence, we only have to approximate the wavevectordependent susceptibility in the disordered phase. We take the trial Hamiltonian of one strip in the disordered regime as follows: „
N
-0H' = -0H + Kth Y, Y cos V(Z^ij - jq) 3 ni'J
+ Ktr]^>2 cos (3Y ^ I J - jq) + cos ( Y^JVJ - jq)
(7)
where H is a restriction of the exact Hamiltonian taking precisely all its terms within the strip, and where 77 denotes the amplitude of the modulated effective boundary fields, h the amplitude of the auxiliary external bulk fields, and q the wavevector of the external field and modulated effective boundary fields along the chiral direction. (In the disordered phase and with a weak field condition, we can expect the response of the spin average to be characterized by the same wavevector q because of the symmetry of H and H'\T)=o,h=o under translation. When we introduce our Weiss and Bethe effective-field approximations, the effective fields should be characterized by this wavevector q as well.) Meanwhile, because most of the previous understanding has come from the study of the Hamiltonian limit, which corresponds to either Kn/Kt -» 0 or Kn/Kt -> oo, 6 " 8 ' 1 2 ' 1 4 it is kind of natural for us to keep Kn/Kt general. Since in all calculations below we take ensemble averages based on H' and often with both 77 and h being zero, we use (• • •) to denote the statistical average with ensemble based on H' and (• • -)o to denote \r)=o,h=o- For convenience, we also define quantities Qc and Qo,j as follows:
Qc
Qd,j
= <
exp ^l-jT«m+l,0j, 1 xp f i — n , J + exp m 0 2
exp (i-|- ni ^)
+ exp
UN = 2m+l, .2TT
(i—nm+ii0J
if N = 2m,
0~JnN'j)
(8)
(9)
These quantities have a direct interpretation: Qc is the spin in the middle row of the 0-th column, if the number of rows N is odd. If the number of rows N is even, we take the average over the two middle rows. Qd,j is the average of the two spins in the boundary rows i = 1 and i = N of the j - t h column (j = —00, • • •, 00). We put the self-consistency conditions (Q c ) = 77
for Weiss approximation,
(10)
(Qc) = (Qd,o)
for Bethe approximation.
(11)
The wavevector-dependent susceptibility has a peak located at qm which gives an approximation to the characteristic wavevector of the corresponding correlation function. By some tedious calculation, the critical point that demarcates the 131
1984 B.-Q. Jin, H. Au-Yang & J. H. H. Perk
paramagnetic-incommensurate phase transition can be located from min ( 1 - Kc ^2(QcQ*d j)oexp(ijq) min
Yl
I = 0
for Weiss approximation, (12)
{(Qd,oQd,j)o ~ (QcQ*d,j}o) exp(ijg) = 0
for Bethe approximation, (13)
3
where the minimum condition is over all q. The corresponding qm will give an approximation to the wavevector qc, characteristic of the correlation function at the phase transition point. Here and in the following we write K = Kt and Kc = Ktc, its value at the critical point separating the disordered and modulated phases. Kn and Kt vary proportionally. In both approximations, the susceptibility near Kc (K < Kc) can be presented in the form
X = */(§-!)• where x 1S * n e coherent anomaly coefficient and cases.21 Prom the above, we can obtain two series of series are short and hence difficult to extrapolate. construct a new extrapolation method as follows. If there exist two sequences {a(n)} and {b(n)},
(14) has been worked out for both approximations. However, both To circumvent this problem, we which satisfy
i) lim n-Hx> cb{n) — c, hnin-^oo bin) — c and a(n), b(n) ^ c for any n, ii) lim„_>oo (a{n + Sn) — a(n))/(b(n + Sn) — b(n)) exists and is not 1, it is possible to construct a third sequence {c(n)} with limn_>oo c(n) = c by a(n + Sn)b(n) — a(n)b(n + Sn) ° W ~ a(n + Sn) - a(n) - b(n + Sn) + b(n)' Under certain conditions, we can expect that the sequence {c(n)} will converge faster than either {a(n)} or {b(n)}. We find that this new construction works very well for the square lattice Ising model and Potts cases with various ratios of Kn/Kt.21 Here we only present the Potts model results, i.e. the case with A = 0 and Kn = Kt, in Table 1, where N Tb Tw Tn
Table 3 1.56208 1.65702 1.5010
1.
Table 4 1.55004 1.62624 1.4992
of T b , T w and T n . 5 6 1.54073 1.53471 1.58794 1.60251 1.4974
7 1.52965 1.57563
critical temperature T\,{N) is obtained by Bethe approximation, TW(7V) by Weiss approximation, N being the width of the finite strip, and Tn(N) is obtained by Eq. (15) with SN = 2. The exact value for N = oo is T* = 1.4925. We clearly 132
The Three-State Chiral Clock Model 1985
see good convergence of series {Tn(7V)}. We have also used this construction to find the range of critical temperatures for A ^ 0. 21 More interestingly, we use this construction to find the characterizing wavevector along the critical line. As is well-known,4 we should be able to get phase transition information through the analysis of the wavevector at the phase transition point. If a Lifshitz point Ai, exists at finite chirality, the characterizing wavevector along the critical line should vanish for A < A/,. Although we are not sure how this Lifshitz point AL will depend on Kn/Kt, old works 4 ' 1 1 ' 1 2 indicate that there is no big dependence of Ax, on Kn/Kt. Two cases with Kn = lO-fQ and Kn = 100.FCt at A = 0.05 have been studied. The results for the two cases are similar, so we only present in Table 2 the results for the case with A = 0.05 and Kn = 100Kt, where the reduced critical wavevector
N
3
0.0463680 0.0362354 -0.00038
Table 2. Table of <Jb, <7w and <jn. 4 5 6 0.0402710 0.0314054 0.00069
0.0353594 0.0276127 0.00099
0.0320544 0.0250293
7 0.0291924 0.0228355
is defined by q = 3q/(2nA). Here, q\>(N) is obtained by Bethe approximation and qv,(N) by Weiss approximation, where N is the width of the finite strip, and qc(N) is obtained by Eq. (15) with 6N = 2. These calculations for the wavevector need an accuracy of 10~ 8 for q. Higher accuracy will be needed for smaller A and our numerical values would not have been reliable enough then. Although we only have three members in this new sequence {qn(N)} and we cannot make a very conclusive case, it looks very tempting to say that this sequence will converge to the true qc from below. Compared with previous results for A^ to be around 0.25 to 0.40, 4 ' 1 1 ' 1 2 we have A t < 0.05. Hence, we may conclude that even for a very small A the wavevector at the transition point is non-zero. This means that the transition should be from the paramagnetic to the incommensurate phase and that possibly no Lifshitz point exists at finite chirality at all.
4. Summary In the above sections, we have used two different methods to approach the problem whether a Lifshitz point exists in the two-dimensional classical three-state chiral clock model at finite non-zero chirality. The first method gives more information about the general phase diagram, whereas the second method seems to more reliably determine the boundary of the disordered phase. However, both extrapolations together consistently indicate that most likely no Lifshitz point exists in this model at finite non-zero chirality. A study of somewhat wider strips on more powerful computers may take away all remaining doubt. 133
1986
B.-Q. Jin, H. Au-Yang & J. H. H. Perk
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
S. Ostlund, Phys. Rev. B24, 398 (1981). D.A. Huse, Phys. Rev. B24, 5180 (1981). J. Yeomans, Physica 127B, 187 (1982). P.M. Duxbury, J. Yeomans, and P.D. Beale, J. Phys. A17, L179 (1984). J.M. Yeomans and B. Derrida, J. Phys. A18, 2343 (1985). P. Centen, V. Rittenberg, and M. Marcu, Nud. Phys. B205[FS5], 585 (1982). G. von Gehlen and V. Rittenberg, Nud. Phys. B230[FS10], 455 (1984). T. Vescan, V. Rittenberg and G. von Gehlen, J. Phys. A19, 1957 (1986). W. Selke and J. Yeomans, Z. Phys. B46, 311 (1982). M.L. Martins and C. Tsallis, J. Phys. A23, 3043 (1990). J.M. Houlrik and S.J.K. Jensen, Phys. Rev. B 3 4 , 325 (1986). S.F. Howes, Phys. Rev. B27, 1762 (1983). S.F. Howes, L.P. Kadanoff and M.P.M. den Nijs, Nud. Phys. B215, 169 (1983). H.U. Everts and H. Roder, J. Phys. A22, 2475 (1989). F.D.M. Haldane, P. Bak and T. Bohr, Phys. Rev. B28, 2743 (1983). H.J. Schulz, Phys. Rev. B28, 2746 (1983). D.A. Huse and M.E. Fisher, Phys. Rev. Lett. 49, 793 (1982). M. Suzuki, J. Phys. Soc. Jpn. 55, 4205 (1986). W.S. McCullough, Phys. Rev. B46, 5084 (1992). B.-Q. Jin, H. Au-Yang and J.H.H. Perk, to be published; see also chapter 2 of http://physics.okstate.edu/perk/papers/jinphd.pdf. 21. B.-Q. Jin, H. Au-Yang and J.H.H. Perk, to be published; see also chapter 3 of http://physics.okstate.edu/perk/papers/jinphd.pdf. 22. F.Y. Wu, Rev. Mod. Phys. 54, 235 (1982).
134
International Journal of Modern Physics B , Vol. 16, Nos. 14 & 15 (2002) 1987-1994 © World Scientific Publishing Company
S T O C H A S T I C D E S C R I P T I O N OF AGGLOMERATION A N D G R O W T H P R O C E S S E S IN GLASSES
R I C H A R D KERNER* Laboratoire de Physique Theorique des Liquides, Universite Paris VI - CNRS URA 7600, Tour 22, 4eme, Boite 142, 4 Place Jussieu, Paris, Prance
Received 24 October 2001 We show how growth by agglomeration can be described by means of algebraic or differential equations which determine t h e evolution of probabilities of various local configurations. T h e minimal fluctuation condition is used to define vitrification. Our methods have been successfully used for t h e description of glass formation.
1.
Introduction
In a series of papers published during the past ten years, 1 ^ new models of growth by agglomeration of smaller units have been elaborated, and applied to many important physical systems, such as quasicrystals, 5 fullerenes,6'7 and oxide and chalcogenide glasses. 8 ~ n Here we shall present the main ideas on which these models are based, and briefly discuss the latest developments. In order to make our presentation concise, the example we choose is the simplest covalent network glass known to physicists, the binary chalcogenide glass AsxSe(i_x), where x is the concentration of arsenic atoms in the basic glass-former, which in this case is pure selenium. The generalization to other covalent networks, e.g. Ge x Se(!_ x ), is quite straightforward. These glasses (in the form of thin and elastic foils) are used in photocopying devices. Whether the formation of a solid network of atoms or molecules occurs in a more or less rapidly cooled liquid, or as vapor condensation on a cold support, the most important common feature of these processes is progressive agglomeration of small and mobile units (which may be just single atoms, or stable molecules, or even small clusters already present in the liquid state) into an infinite stable network, whose topology can no longer be modified unless the temperature is raised again, leading to the inverse (melting or evaporation) process. To describe such an agglomeration with all geometrical and physical parameters, such as bond angles and lengths, and the corresponding chemical and mechanical *Email: [email protected] 135
1988
R.
Kerner
energies stored in each newly formed bond, is beyond the possibilities of any reasonable model. This is why stochastic theory is an ideal tool for the description of random agglomeration and growth processes. Instead of reconstructing all local configurations, it takes into account only the probabilities of them being found in the network, and then the probabilities of higher order, corresponding to local correlations. This is achieved by using the stochastic matrix technique. A stochastic matrix M represents an operator transforming given finite distribution of probabilities , \pi,P2, —,PN] > into another distribution of probabilities, \pi,p2, ...,pN]. It follows immediately that such a matrix must have only real non-negative entries, each column summing up to 1. The algebraic properties of such matrices are very well known. The main feature that we shall use here is the fact that any stochastic matrix has at least one eigenvalue equal to 1. The remaining eigenvalues have their absolute values always less than 1. This means that if we continue to apply a stochastic matrix to any initial probability distribution, after some time only the distribition corresponding to the unit eigenvalue will remain, all other contributions shrinking exponentially. This enables us to find the asymptotic probability distribution. In what follows, we identify these probability distributions with stable or metastable states of the system, fixing the statistics of characteristic sites in the network. Taking into account Boltzmann factors (with chemical potentials responsible for the formation of bonds), we are able to find the glass transition temperature in various compounds. In particular, one is able to predict the initial slope of the curve Tg(c), i.e. the value of (dTg/dc)c=0.12'13 2. Stochastic matrix describing cluster agglomeration Consider a binary selenium-arsenic glass, in which selenium is the basis glass former, and arsenic is added as modifier (although its concentration can be as high as 30%). The chemical formula denoting this compound is AscSe^c), where c is the As concentration. In a hot liquid, prior to solidification, the basic building blocks that agglomerate are just selenium and arsenic atoms, indicated respectively by (—o—) and f—• J. When the temperature goes down, clusters of atoms start to appear everywhere, growing by agglomeration of new atoms on their rim. Consider a growing cluster: one can distinguish three types of situations (we shall call them "sites") on the cluster's rim. The concentration of free As atoms in the liquid will be called c and that of Se, (1 — c). Two choices are possible for constructing the states and transition matrix (see Ref. 14). There are three possible kinds of sites: a selenium atom with one unsaturated bond, and an As atom presenting one or two free bonds; these are indicated by x = o—, y = * and z = •—. To each site one of the two basic cells can attach itself, reproducing one of the initial configurations, in the specific combinations shown in the next column of the Figure 1. The attachment of one single basic cell, or the saturation of one single bond, is a step in the evolution. In the second choice, 136
Stochastic Description
of Agglomeration
and Growth Processes
1989
each step is obtained by the complete saturation of all the bonds at the rim, so that only two types of sites (denoted by x and y) are seen on cluster's rim, assuming that the growth is of dendritic type (no small rings present). It can be shown 14 that the two approaches lead to the same results, which may be considered as a proof of the ergodicity of the proposed model. We shall choose the second version of the model for the sake of simplicity. In this case, we can take into account only the x and y-type sites, because the z-type sites transform after the next agglomeration step into an x or y type site. The elementary step in the agglomeration process, described by he transition matrix, corresponds now to the complete saturation of all the available free bonds on the rim. This is represented in Figure 1 : 2 (1 - c) e~£
x
Zee'71
V
-A
y
4 ( 1 - c ) 2 e - -27} 12 c ( l - - c) e"-n-
2x x —<<~ +y
9 c2 e-2a
2y
Figure 1: States, steps and un-normalized probability factors . Observing that from the site z only the sites of x and y type can be produced, we can forget it and consider the dendritic growth with only two types of sites appearing all the time. Given an arbitrary initial state {px,Py), the new state results from taking into account all possible ways of saturating the bonds of the previous state's sites by the available external atoms. The un-normalized probability factors are displayed in the Figure. The non-normalized probability factors can be arranged in a matrix 2(l-c)e-£ 4(l-c)2e-2" 8(1 - c) 2 e- 2 " + 12c(l - c)e'r>-a 12c(l - c ) e - " - a + 1 8 c 2 e - 2 a J ^ The normalized transition matrix is written as MXy\ = / MXX M=(MXX \MyX Myy) \1-MXX
1-Myy\ Myy )
. ^
where the entries are obtained by normalizing the columns of the matrix (1). 2(1 - c)j Mxx = „ , _ , / , * _ , 2(l-c)£ + 3c'
and
3c/x Myy 7 , o_ w = „,,, " 2 ( l - c ) + 3cM
(3)
where we have introduced the abbreviated notation £ = ev~e and \i = ev~a. The eigenvalues of this matrix are 1 and Mxx — Myy = Mxy — Myx. and the stationary eigenvector is p?\_ P^ J
1 (Mxy Mxy + Myx VM, 137
(4)
1990
R.
Kerner
It can be seen from Figure 1 that on the surface of an average cluster, px is the Se concentration and py is the As concentration. Now, the high homogeneity exhibited by known glass structures suggests that even in relatively small clusters, deviations from the average modifier concentration c must be negligible. Thus, in the bulk, the As concentration should be equal to c. Therefore, the condition of minimal fluctuations in the bulk concentration can be interpreted as the glass transition condition. This means that the asymptotic state is fixed by the external concentration, therefore the above eigenvector must be equal to the average distribution vector (1 — c, c). The solutions are c = 0, c = 1 and the nontrivial one Myx Mxy + Myx
6 - 4£ 12 - 4£ - 9/J,
(5)
This equation can be checked against experiment. For example, we can evaluate the derivative ^ = (^f) for a given value of c. In particular, as c —> 0, we can neglect the As-As bond creation (equivalent to putting /i = 0 in (5)), to get dT dc c=0
ln(3/2) '
(where Tgo is the glass transition temperature of pure Se). This is the present-case expression of the general formula given by the stochastic approach, the fraction (3/2) being replaced by (m'/m), where m and m' are the valences of the basic glass former and of the modifier), remaining in very good agreement with the experimental data (see Refs. 15-17). 3. Low concentration limit. The above scheme can be easily generalized to the case of arbitrary valence, say rriA and TUB- In that case, the stochastic 2 x 2 matrix has the same form as (2), but with the entries given by
Mxx = l-Myxvx =
I?A(1"C)*
,
Mxyxy = l-Myyvy=
m,i(l-c)£ + mBc'
mA(l
"
A ( 1
- c)
"
C )
+mBcn
»
The asymptotic probability has the same form as before, as well as the zero fluctuation condition relating c with T (interpreted as the glass transition temperature). The derivative of c with respect to the temperature T gives the "magic formula"
dc
1 (%£-pKK-(%f-0^
dT
T
[(1-H^) + ( 1 - ^ ) ]
2
(6)
where we used the fact that j | ; = — ^(,ln£, , and -^ = — ^ /j,lnfi. This defines the slope of the function Tg(c), which is an important measurable quantity :
138
Stochastic Description of Agglomeration and Growth Processes
1991
The initial slope, at c = 0, is of particular interest. Its expression is very simple, taking into account that when c = 0, we have also £ — ^ - , which leads to dTg dc Its value has been checked against the experiment very successfully, in more than 30 different compounds. In some cases the formula does not seem to work well; usually it comes from the change of valence of certain atoms provoked by the influence of the surrounding substrate. One could be worried about the apparent singularity in this formula when TUA = TUB, i.e. when one deals with a mixture of two different glass formers with the same coordination number. It is not difficult to show that also in such a case a reasonable limit can be defined, as has been recently suggested by M. Micoulaut. 19 As a matter of fact, suppose that the glass transition temperature of the pure glass-former A is Tgo, and that of the pure glass-former B is Tg\. We can re-write our minimal fluctuation condition in a very symmetric manner, invariant with respect to the simultaneous substitution JUA *-> m s , c «-» (1 — c) and £ «-> /x : C
(l-c)[(l-c)(l-^O-c(l-^M)]=0 TUB
(9)
TTIA
Obviously, the "pure states" c = 0 or c = 1 represent stationary solutions of (9) and can be factorized out. The non-trivial condition for the glass forming is thus (l-
C
)[l-2±0-c[l-3£ TUB
M
]=O
(10)
THA
Now, using the limit conditions at c -¥ 0, Tg = Tg0 and c -> 1, Tg = Tgi, and introducing the generalized Boltzmann factors with the energy barriers for corresponding bond creations as EAA, EAB and EBB, we can write EAB-EAA = kTgQln{—), EAB - EBB = kTglln^), (11) m,A TUB so that the expressions £ and fi at the arbitrary temperature T can be written as E
£(T) = e
E AR~ AA T
*°
T
gO
T
771 R
Tg0
= p £ ) + ;
B
n{T) = e
TTlA
E AB~ BB T
»
Zsl
^
171, A
= (l2±)+.
T
sl
(12)
TUB
Substituting these expressions into (7) and taking the limit c ->• 0, we get dTqi
Tg0[l-(^Y
dc >"-"
ln(**) TTIA '
(13)
It is easy to see now that even when TUA = TUB, this formula has a well defined limit. Indeed, if we first set ^ = 1 + e, and then develop the numerator and the denominator of the above equation in powers of e, then in the limit when £ - > 0 w e arrive at a simple linear dependence which is in agreement with common sense and with experiment as well, namely f
|c=o=T9i-rfl„ 139
(14)
1992
R.
Kerner
This formula is also confirmed by many experiments, e.g. performed on seleniumsulfur mixtures (where VHA = TUB = 2). The deviations from the linear law (14) observed in the Se — Te binary glass are explained by the fact of the chemical properties of tellurium, which changes its valence from 2 to 3 in presence of selenium. 4. The effect of rapid cooling An interesting extension of this model is obtained when we take into account the effects of rapid cooling, i.e. when the time derivative of the temperature can no longer be neglected. The treatment of this problem was suggested in Ref. 20, and has been solved quite recently. 21 Consider the agglomeration process defined by the above stochastic matrix, p' = Mp, with p representing a normalized column (a "vector") with two entries, px and py = 1 — px- After one agglomeration step, representing on the average one new layer formed on the rim of a cluster, we can write Ap = p ' - p = ( M - l ) p
(15)
Let us introduce a symbolic variable s defining the progress of the agglomeration process; obviously, s(t) should be a monotonically increasing function during the glass transition. If the temperature variation is so slow that the derivative dT/dt = (dT/ds)(ds/dt) can be neglected (which is often called the annealing of glass), the master equation of our model can be written as Ap = ~ As = (M -
l)pAs
OS
where the variation As represents one complete agglomeration step. If we want to use real time t as an independent parameter, we should write ^ = A p d f = r _ 1 A p = l Jy y dt Asdt As ry ' x We have introduced here the new entity r = (ds/dt)~ which can be interpreted as the average time needed to complete a new layer in any cluster, or alternatively, the time needed for an average bond creation. Now, if the temperature varies rapidly enough, the matrix M can no longer be considered as constant. The equation (16) must be modified according to the well known "moving target" principle. That is, the total derivative of p with respect to t should read: dp ,,r ^,ds - = (M-l)-p+
_
dM dT _ dp — - P = -
-{M-l)+q
— P
(17)
where we supposed linear dependence of the temperature on time, so that the derivative dT/dt can be denoted by constant cooling rate q. In the two-dimensional case only one component of pis independent, because px +py = 1. Let us choose py (whose asymptotic value should be equal to c) as independent variable. Then (17) will reduce to the single equation : dpy _ 1 (Myy-l)py+Myx(l-Py) (18) Py) + q dT Py+ dT u dt 140
Stochastic Description of Agglomeration and Growth Processes 1993
where we have used the fact that px = 1 — py , Mxx = 1 - Myx and Myy = 1 — M x y . What remains is just simple algebra. After a few operations we find the asymptotic value of py, denoted p^, obtained when we set dpy/dt = 0 : Myx+rq(9-^)
p0O = Py
(Mxy
+
Myx)+rq(d^^M^)
^
As in the former case, we define the glass transition temperature by solving the zerofluctuation condition p^ = c. The quasi-equilibrium condition thus obtained can be written in a form displaying an apparent symmetry between the two ingredients ("A" and "B") of binary glass. As in the previous case (when q = 0), the limit values c = 0 and c = 1 represent stationary solutions, which is obvious (no local fluctuations of concentration c are possible when there is no ingredient other than AOT B atoms alone). After factorizing out c(l — c), we get mB mA(l
rq —
mAmB
mA
— c)£ + THBC
mA(l - C) + TTIBCH
c^iln/j, [mA(l - c) + mBcfi}2
(l — c)£ln£ [mA(l - c) f + mBc]2
(20)
where we have used the fact that ^ " 4 * = - ^ , ^ ^ = - ^ . The above formula seems quite cumbersome, but it become much simpler in the low concentration limit, c —> 0 Close to c — 0 we get ^-S+^lnt =0 (21) mA T mA (quite obviously, in the limit c —> 1 one gets the same formula switching mA with TUB and replacing £ by /z). Replacing £ by the expression (12), we arrive at :
fmB\'
\mAJ
+
<¥>^"0=°-
<22>
It is easy to see that independently of the ratio TUB/'mA, for temperatures T above Tgo we must have q < 0, and vice-versa, during rapid cooling the glass transition occurs at the temperature T > TgoThe dimensionless combination (rq)/T defines the quenching rate as the product of (l/T)(dT/dt) = d{lnT)/dt by the time constant r, characterizing the kinetics of the agglomeration process, i.e. the average time it takes to create a new bond. It may depend weakly on the temperature, but for the sake of simplicity suppose it is constant. It can be determined by comparing formula (22) with the experimental data. To take an example, let us again consider the selenium-arsenic glass at c —> 0 (almost pure selenium with a small addition of As). We know that in this case Tg —> Tgo = 318°K. The formula (22) then gives the quasi-linear dependence of A T = T - Tg0 on the quenching rate q: for Tg = 328°^ (i.e. A T = 1Q°K) we get rq = -10.38; for Tg = 338°i<: (i.e. AT = 20°if) we get rq = -21.51; for Tg = 348°ii: (i.e. AT = 3 0 ° ^ ) we get rq = -32.26, and so forth. 141
1994 R. Kerner Finally, if we want t o establish the formula for a p u r e glass-former, without any modifier, we should t a k e t h e limit (ruA/ms) -> 1 and fi - 4 £; we t h e n get ^
^
+ (^)^=0
or
T-T0
= ATg = -(rq)^f.
(23)
Eventually, the deviations from this simple dependence m a y indicate t h a t t h e characteristic time r depends on T. This can shed more light o n t h e agglomeration kinetics in various glass-forming liquids. More details can b e found in Refs. 18,21. Acknowledgements Enlightening discussions with R. Aldrovandi, R.A. Barrio a n d M. Micoulaut are gratefully acknowledged. References 1. R. Kerner, Physica B 2 1 5 , 267 (1995). 2. R. Kerner and M. Micoulaut, Journ. of Physics: Cond. Matter, 9, 2551-2562 (1997). 3. R. Kerner, A theory of glass formation, in Atomic diffusion in amorphous solids, M. Balkanski and R.J. Elliott, eds., World Scientific, pp. 25-80 (1998). 4. R. Kerner, The principle of self-similarity and its applications to the description of non-crystalline matter, Proceedings of the Workshop in Cocoyoc, 1997, J.L. MoranLopez ed., Plenum Press, pp. 323-337 (1998). 5. D.M. dos Santos-Loff and R. Kerner, Journal de Physique I (4), 1491-1511 (1994). 6. R. Kerner, K.H. Bennemann, K. Penson, Europhysics Lett. 19 (5), 363-368 (1992). 7. R. Kerner, Computational Materials Science, 2, 500-508 (1994). 8. D.M. dos-Santos-Loff, R. Kerner and M. Micoulaut, Journ. of Phys. C, 7, 8035-8052 (1995). 9. R.A. Barrio, J.P. Duruisseau and R. Kerner, Phil. Magazine B, 72 (5), 535-550 (1995). 10. R.A. Barrio, R. Kerner, M. Micoulaut, G.G. Naumis, Journ. of Phys.: Condensed Matter, 9, 9219-9234 (1997). 11. R. Kerner, G.G. Naumis, Journal of Non-Cryst. Solids 231 111-119 (1998) and R. Kerner, G.G. Naumis, Journ. of Physics: Cond. Matter 12 (8), 1641-1648 (2000). 12. R. Kerner, Journ. of Non-Cryst. Solids, 182, 9-21 (1995). 13. R. Kerner, M. Micoulaut, Journ. of Non-Cryst. Solids, 210, 298-305 (1997). 14. R. Aldrovandi and R. Kerner, in "New Symmetries and Integrable Models," Proceedings of XIV Max Born Symposium, eds. A. Prydryszak, J. Lukierski and Z. Popowicz, World Scientific, pp. 153-169 (2000). 15. P. Boolchand and W.J. Bresser, Phil. Mag. B 80, 1757-1772 (2000). 16. P. Boolchand, X. Feng and W.J. Bresser, Journ. of Non-Cryst. Solids, 293-295, 348356 (2001). 17. D.G. Georgiev, P. Boolchand and M. Micoulaut, Phys. Rev. B 62, R9228 (2000). 18. R. Kerner, Mathematical models of glass formation, Proceedings of the conference "Glasses and Solid Electrolytes" (St. Petersburg, May 1999), Glass Physics and Chemistry, 26 (4), pp. 313-324 (2000). 19. M. Micoulaut, private communication (2001). 20. R. Kerner, M. Micoulaut, C.R. Acad. Sci. Paris, t. 315, Ser. II, 1307-1313 (1992). 21. R. Kerner and J.C. Phillips, Solid State Communications, 117, 47-56 (2000); R. Kerner, in preparation.
142
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 1995-2001 © World Scientific Publishing Company
FUSION C O N S T R U C T I O N OF T H E V E R T E X O P E R A T O R S IN HIGHER LEVEL R E P R E S E N T A T I O N OF T H E ELLIPTIC Q U A N T U M GROUP*
HITOSHI KONNO Department of Mathematics, Hiroshima University, Higashi-Hiroshima 739-8521, Japan E-mail: [email protected]
Received 15 November 2001 After a short summary on the elliptic quantum group Bq:\(sl2
) and the elliptic algebra
UqlP(5l2), we present a free field representation of t h e Drinfeld currents and the vertex operators (VO's) in the level k. We especially demonstrate a construction of the higher spin type I VO's by fusing t h e spin 1/2 type I VO's and fix a rule of attaching the screening current S{z) associated with t h e g-deformed Z^-parafermion theory. As a result we get a free field representation of t h e higher spin type I VO's which commutation relation by the fused Boltzmann weight coefficients is manifest.
1. Elliptic Quantum Group and Elliptic Algebra 1.1. Elliptic
Quantum
Group Bq<\(sl2)
[2]
The face type elliptic quantum group Bq:\(sl2) is a quasi-Hopf algebra (Bq,\(sl2), A\, e, S, 3>(A), a, /3, 7£(A)) obtained as a deformation of the Hopf algebra (Uq(sl2), A, e, S, 11) by the face type twistor F(X) (A G F)) satisfying the shifted cocycle conditiona F (12) (A)(A ® id)F(A) = F<23>(A + h ^ ) ( i d ® A)F(A).
(1.1)
The "deformation" means that Bq,\(sl2) = Uq(sl2) as an associative algebra, but the coalgebra structure is deformed in the sense A A ( ^ ) = F(X)A(x)F~1(X) Vx e 21 12 1 Uq(sl2), K(\) = F( )(A)'RF( )- (A),etc. The universal R matrix 1l(X) satisfies the dynamical Yang-Baxter equation. ^(12) (A + h&)U^
(A)7^23> (A + h™) = 11^
(X)Tl^
(A + h^)K,W
(A). (1.2)
^From this, we obtain the dynamical RLL relation which characterizes Bqt\(sl2) R^w(z1/z2,
X + h)L+(zuX)L+r(z2, = L+,(z2, X)L+(zi,X
X+
+ hW)R+w(Zl/z2,
*A short report on a part of the work done with Robert Weston. We follow the notation in Ref. 2.
a
143
h^) A).
(1.3)
1996
H. Konno
Here L$(z,\) = (irv,z ® id) 11+(\), R^w(z1/z2,X) = {-KV,ZI ® nw,Z2)K+{\), + c d+d c K {\) = q ® ® 1l(\) in the evaluation representation {TVV,Z,VZ). Hereafter we use the parameterization A = (r — c + 2)d + (P + 1)5/11 and A + /i = (r + 2)d + ( P + /ii + l ) | / t i (fti G fj). Then the two dimensional representation {^v^.zi ® 7rv<1),z2) ° ^ t n e -^ matrix, up to gauge transformation, is given by
R+{z, P+hi)=
6(u,P + /ii) c(u,P + fti) c ( u , P + /ii) 6(u,P + /ii)
i # ( 1 ) v ( 1 ) (z, A + h) = p+(«)
V
1/
P*+(2:, P ) = P+ ( 1 ) v ( 1 ) (z, A) = R+(z,P)\r Here z = q2u, p = q2r, r*=r-c,p*
= q2r* and p+(u) = ^
{z} = {z;P^U
}
and 6(«,5) = ^
^
{
^ffz}
<'gffp,
\ c(u,s) = { f g | , c(u,s) =
b(u, s) = J " L . The symbol [u] denotes the Jacobi theta function [u] — q^ 1 ;!!
fag§, " /^.5a ,
= tap)oo(p/*;p)oo(p;p)oo, (*;PI, •••,?*)«> = rin 1 ,...,n»>o( - Pi "'Pfc £ 3We also use [u]* = [u] |r->.r«. These R matrices are nothing but the Boltzmann weight of the Andrews-Baxter-Forrester (ABF) model. Hence one can regard Bqi\{s\2) as a central extension of Felder's elliptic quantum group EV!7,(s[2). ©P(Z)
1.2. The elliptic
1
algebra L/q)P(sl2)
The algebra Uq>p(sl2) is an elliptic analogue of the algebra Uq(sl2) in the formulation via the Drinfeld currents. Our currents E(u), F(u), K(u) satisfy the following relations. 3 K(u)K(v)
= p(u \U — V
v)K(v)K(u), 1—r* 1*
"(«W») = [,,,,,_ iyj,E(»)g(»).
[u-v[u-v
l]*E(u)E(v) + l]F(u)F(v)
Here p(u) = p+*(u)/p+(u),
= [u-v + l}*E(v)E(u), = [u-v-
l]F(v)F(u),
p+*(u) = p+(u)\r^rr*
and
_ ^ ( Z ' P * ' 9 ) ^ r . _ nx _ (92z;p,g4)oc(pg2^;p,g4)oo witn/t _— liT 11m — r , ?(,2,p,q — aiz.„ai\—u,-nai\—• witVlK
144
Fusion Construction of the Vertex Operators 1997
Let us introduce the half currents K+(u),
E+(u) and F+(u) by
K+(u):=K(u+^),
F+(u):=a
Jc 2mz>
(1.4)
F ( u ' - -)-, y 2>[u-u'
f^ + \][P + h1-l]
(1.6) y '
Here C* : \p*q~lz\ < \z'\ < |g -1 ;z|, C : \pqz\ < \z'\ < \qz\. a and a* are the normalization constants satisfying °_JJi = 1Theorem 1.1: Define the L-operator L+(z) € End Vz(1) ® Uq
-~\o
i JV o
0 +
\(
1 0 +
jr (u)-vl,£ (u)i
Then we have the following i?L-L-relation. R+ (zx/za, P +fci)£ + (zi)L+(z 2 ) = L + ( z 2 ) £ + ( z i ) i T + (zj/za, P).
(1.7)
^Prom this, one can recover the dynamical RLL relation (1.3) in V = W = V^ by /e-Q o \ introducing the new -L-operator L+(z,P) := L+(z) I Q I . In fact, t/g, p (sl 2 ) can be regarded as an extension of the algebra Bqt\(s\2) by an extra element e® and imposing the commutation relation [P, e®] = —e Q . 4 The elliptic algebra £/qiP(sl2) hence provides an alternative formulation of Bqt\(sh) via the Drinfeld currents. 1.3.
The vertex
operators
of
UqiP(sl2)
^,From the spin 1/2 type I and II intertwining operators $yff(z), Uq(sl2), we define the Bq^siz) 1/2 as follows.
$*y{%'(z) of
intertwiners ^ ^ y (z, A) and **^ ( ',y(z, A) of spin
^ ( z , A) := (id ® 7r„<„,jF(A) o ^ ( z )
: ^ ) - > V{v) ® y<'\
* * ^ ( z , A) :=tt*J#>(z) o (7r v(0iZ 0 id)F(A)" 1 : V® ® F(/,) - » • * » . Here V(//) is the level k highest weight t/g(s[ 2 ) module with the highest weight \x and Vz is the I + 1 dimensional evaluation representation. Furthermore, we can extend them to the spin 1/2 VO's acting on the Uq>p(sl2) modules V(/x) := ® n g Z F(/i)(g>enQ as follows. *W»<»)(z) := ^ ( z , A) : t?(/x) - • V > ) ® V<'\ **(*)(",/*)(s)
:=
* ^ (
z >
A)e hl ®° : V ^ ® f (p) ->• ?(i/). 145
(1.8) (1.9)
1998
H. Konno
Then from the intertwining relations of $y$(z, A) and ^y^\ their analogue for the VO's of Uq^sh) as follows. $(')(-.M)( U ,)L+( Z )
= i2+( 1 ) v ( i ) ( ^ / U ; ) P
£+(z)r(m^)(w)
=
^Km^)
W
£
(z, A), we can derive
+ /ll)L+(z)$W(^)(w)) + w
^+)^
) ( z M
(i.io)
p_^(i)_^)
) ( L 1 1 )
It is also possible to derive the commutation relations of the VO's. Especially, the spin 1/2 type I VO's satisfy
R{z,P + hl)^v^\zl)^K\z2)
K
= ^2^v^(z2)^,i''K\z1)W'
,
/i
zi/z2
(1.12) where R(z,s) — PR(z,s), W" = W\r^k+2.
R(z,s)
+
+
= R (z,s)p(u)/p (u),
Pa®b
= 6 ® a and
2. Free Field Representation The CFT limit of the ABF model and its fusion models are described by the coset {sh)k ® (sh)r-k-2/{sh)T-2 Virasoro minimal model, which is known to be equivalent to the tensor product of the Z^-parafermion theory and one boson theory with a certain background charge. Corresponding to this fact, the elliptic algebra ^q,p(sb) can be realized by using the (/-deformed Z^-parafermion theory and one (/-boson theory. 2.1. Drinfeld
currents [3J
We use three kinds of bosons satisfying the relations. [2n][fen] H n [r*n\ [2n][(fc + 2)n] [ai,n,ai,m\
r
[d2,n,a2,m\
=0n+m,0
1
x = ~ <Wm,0
2kr r* ,
["l,Qll —2{k
+
2),
n [2n][kn] ,
1^2, W2J = —*•&.
We also set a'0n = t ^ f ao,n- As usual, it is convenient to introduce the corresponding boson fields. UM
B, C\z; D) = - A ( Q , + P. log.) + £
J^
m 1
<^f ) (A;i?|z;C') = § l o g g + ( g - Q - 1 ) ^ | 4 4 ^ ± ™ ^ ^
m>0
Wm
3
m
C m
g
D | m |
>
C? = 0,1,2)
^ ^
and 0d(^4;B,C|z) = fo{A\B,C\z)\r->r»,a0jn^,ai . We often use the abridgment ^(C|;?;Z>) = ^(i4;i4,C|z;I>), fa(C\z) = ^(C\z-0) etc. 146
Fusion Construction of the Vertex Operators 1999 Now let us define the ^-analogue of the Zfc-parafermion fields $(z) and ^ ( z ) by $(z) = * _ ( z ) , &(z) = * + ( z ) with
**<*) = T(q-q-i)
: ex
p{±^(c|z; ± | ) }
x(exP{-4+)(l;2|,;T^)±#)(l;2|,;T0} -e,P[4-\l^T^)T4-\l;2\z^l)}):. The Zfc-parafermion theory contains the screening current S(z), which commutes with ^±(z) up to the total difference. S(z) = —
^ : exp{fc (k + 2\z; -
x(exp{4
+ )
(l;2|,;^)
^ +
) } ^(l;2|,^)}
-«xp{-^-)(l;2|,;*±^)-^(l;2|z^)}):. Then we have, Proposition 2.1: The Drinfeld currents ofUqyP(sl2) K(u) = z~*& e-*i(i;2,r*|*)>
#(u)
=
at c = k are given by
^ ) e-toW*),
p(u) = *\z) e ^ ^ .
We regard these currents as the operators acting on the following Fock spaces. M
TP3FM
= C[au,a2J
(I G Z< 0 )] <8> e W
Q l
® e&Q\
Tt% = C[a0,i (Z G Z< 0 )] ® e ^ £ Q » - « ° , where a a , m = ^ p a - + ^ a + , V^-)
«-
J~a-2,m;Ji
=
-\/if-
»S'(z)
:
I n
faCt
'
{I < a < r — 1, 1 < m < r — A; — 1), a + =
- B '( Z )
:
Fa,m;J
->
Fa,m-2;J,
F(z)
: Ta,m.tj
->
3~a,m;J ~^ 3~a,m;J-2-
2.2. Tfte type / vertex
operators
2.2.1. 77ie aptn-1/2 W ' s Let iij = (k — J)A 0 + JAi be the level k sl 2 weight. We set J' = J + a (a = ±1) and define the components of the spin 1/2 VO as follows. $ ( / W ) ( U ) = gW>J)(z) ^
*?(*) ® Ve,
(2-13)
£=±1
where {ve}e=± is a basis of the two dimensional representation V^. We realize $"(z) as an operator *°(z) : ^ a ,m;J ->• Ta-e 147
2000
H. Konno
In the case a = + , solving the "intertwining relation" (1.10) with / = 1, we obtain 5>+(z), after a certain gauge transformation, as follows.4 $
" ( 2 ) = W]MZ) : e"*i(li2,*l*) :>
( 2 - 14 )
[u U + + = F+(u)*±(z) = I £- ' ^ K]F{z')*t{z). (2.15) 1 JCpz 2mz' [u-u - ±] Here we set K = P + h\. i,i(z) is the q-analogue of the spin 1/2 parafermion primary fields. In general, the sipn 1/2 (I = 0,1,..., k) field (pi,i{z) is given by
*J(z)
4>i,i(z) = -expi
-<j>2 M;2,fcz;£)
- 0 i (Z;2,fc + 2
^
H = . (2.16)
The VO's &£(z) satisfy the following commutation relation.
With p'(u) = p(u)\r-*k+2On the other hand, the VO's $7(z) require an attachment of the Zfc-parafermion screening current S(z), because among E(z), F(z), S(z), only S(z) can decrease the quantum number J. However since the g-deformed Z^-parafermion theory has no coalgebra structure, we have no definite guiding principle to fix the rule of S(z)attachment. We do this by hand for the spin 1/2 VO's requiring the commutation relation (1-12) and extend it to the higher spin VO's by fusion procedure. We thus obtain the following realization of the spin 1/2 VO's. dw
*+r \ot
s\u ~ v ~ \ ~
2
JCs * ™W
\u-V-
p
i\'
5'
Here [it]' = [u]\q2r_>.q2(k+2) and the contour Cs,z should be chosen in such a way that the poles z' = qk+1 Zq2(k+2)1 (I = 0,1,2,..) are inside whereas the poles Z' = q-h-lzq-2(k+2)l
y
= 0 > j^ 2 j
)a
r e o u t g i d e
2.2.2. Fusion construction of the higher spin VO's We next consider the fusion of the I spin 1/2 VO's. For a — Ylj=i a j>
*W°(^'- 1 )= J2 *? 1 1 (V ( '- 1) )^(^ 2( '- 2) )-"*?, , W.
w e set
(2-18)
It turns out that the simplest component $_] (z) is expressed by using the spin 1/2 <7-parafermion primary field
^{zq1-1)
= $t(^-1))$i(^-2))...$±(z)
148
Fusion Construction of the Vertex Operators 2001 with C(z) being a function appearing from the normal ordering. Lemma 2.2: In (2.18), let j a (1 < a < m(< I)) be integers satisfying ctja = —, and for other j , aj = +. Then in (2.18) one can move all S(z) in ~ (z) to the right and obtains the expression which has no j a (a = 1, ..,m) dependence at all.
l 1) 2{l 2) ^K(^ )^ 2(zQ - )--^tM)(f *!,..,*,
ycs.z
dmSiw^-^^A lu v
2J
/
.
This indicates that the VO's 3>£ (z) have no intermediate-weight-path dependence. Hence $£ (z) manifestly satisfies the commutation relation for the spin 1/2 VO's.
J2Rii(zi/z2,K)i^^(z1)^^(z2) £l,E2
where II = Pi + 1, Ru and W'u are the I x I fused ABF Boltzmann weights. Acknowledgments The author would like to thank Michio Jimbo, Satoru Odake, Atsushi Nakayashiki, Jun'ichi Shiraishi and Robert Weston for valuable discussions. References 1. 2. 3. 4.
H. Konno and R. Weston, in preparation. M. Jimbo, H. Konno, S. Odake and J. Shiraishi, Transform. Groups 4, 303-327 (1999). H. Konno, Comm. Math. Phys. 195, 373-403 (1998). M. Jimbo, H. Konno, S. Odake and J. Shiraishi, Comm. Math. Phys. 199, 605-647 (1999).
149
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2003-2008 © World Scientific Publishing Company
Q U A N T U M DYNAMICS AND RANDOM MATRIX THEORY HERVE KUNZ* Institute of theoretical Physics, Swiss Federal Institute of Technology Lausanne, PUB - Ecublens Lausanne , CH-1015 EPFL, Switzerland
Received 15 November 2001
We compute the survival probability of an initial state, with an energy in a certain window, by means of random matrix theory. We determine its probability distribution and show t h a t is is universal, i.e. characterised only by the symmetry class of the hamiltonian and independent of the initial state.
In classical mechanics, temporal chaos is characterised by the extreme sensibility of a trajectory to variation of initial conditions. No direct analog of this phenomenon has been found in quantum mechanics so far. On the other hand, numerical evidence has been accumulated, 1 showing that energy levels of a quantum system, whose classical counterpart is chaotic, have a statistical behavior described by Wigner's random matrix theory (RMT), on the mean level spacing scale. The question we want to address is the following: are there specific predictions of RMT for quantum dynamics, which would characterise the temporal behavior of " chaotic" quantum systems. We consider the following situation: The system is prepared in an initial state ip at time 0, with an energy in a certain window, centered at e and of width 2sl(e), where 1(e) is the mean level spacing, and we want to compute the probability to find our system again in the state tp, at a later time t. This quantity that we call the survival probability R is given by (ylexpifjH
P(A)
iff, P(*)
(1)
H is the hamiltonian of our system and P(A) is the spectral projector on A. We have chosen to take an energy in a range of the order of the mean level spacing in order to look at properties of the system which are independent of specific details. If *[email protected] 151
2004
H. Kunz
(\j, ipj) denote respectively the jfh eigenvalue and eigenvector of the Hamiltonian H, then the survival probability can be written as R =
E,=i
VjXj^P^irxj
E»-i
©
Z J , = I VjXj
(2)
where (3)
Vj = l(v. Vv)l and if we define Xj by the relation Aj — e + Xjl(e)
(4)
J01 *(if | a;j | < s
( \
Xj=X(-^)(2;j) = |
otherwise
(5)
The Heaviside function 0 ensures that there is at least one eigenvalue in A. What appears naturally in this expression is the time measured in units of the Heisenberg time iff =
h 1(e)
(6)
so that t
(7)
iff"'
If we look at this problem from the point of view of RMT, we will replace the Hamiltonian by a large N x N self-adjoint matrix, whose probability distribution is basis independent and therefore of the form e-w(x^..,xN)
dH
(8)
Wigner's gaussian model corresponds to the choice N
(9)
2
The first conclusion to be drawn is that the survival probability is statistically independent of the initial state ip. This follows from the fact that the variables {Vj}j=1 have a probability distribution, independent of ip and given by: 1
I N
\
m(y)dy = -g- s \J2 yj -1 \j=l
/
N
n yJ'1 dy
(10)
3=1
The parameter /3 = 1, 2, 4 characterise the symmetry class of the Hamiltonian, respectively orthogonal, unitary and symplectic. Equation (10) follows easily from the Haar measure on the corresponding groups. CN is a normalising constant. 152
Quantum
Dynamics
and Random
Matrix Theory
2005
The variables {XJ}.=1 are statistically independent of the variables {yj} . = 1 and have a distribution given by •^-exp-W(e + xl(e))A0(x)dx
(11)
where the Van der Monde determinant A(x)=
J]
\xi-Xj\
(12)
l
comes from the change of variables Hij —¥ (Xj, "0j) 7=1 - 2 DN is a constant of normalisation. We can take l{e) = ]y~j^y, where p(e) is the density of states when N = oo. The problem that we need to solve now is to find the probability distribution of the survival probability p(R)dR in the N = oo limit. We find that R is not selfaveraging i.e. p(R) is not a delta distribution concentrated on the mean value of R. On the other hand its probability distribution p(R) is universal, i.e. it depends only on the symmetry parameter (5, at least for a large class of W. There are two formulas for p(R), one more appropriate to small windows, another one to large windows. In the first case, we decompose p(R) into p(R\r) = J2r^¥pn(R\r)
(13)
71=1
where En is the probability to find exactly n eigenvalues in A and pn(R\r) is the conditional probability density of R knowing that there are exactly n eigenvalues in A. It can be expressed as 2>
Pn{R\r)= I \\E(xi,...,xn)dnx
fin(zi,...
,z„)dnzS
-
N
Zj e x p 2TTITX3
y (14) E(Xl,...,xn)=E{xl--'Xn)
(15)
E(Xl,...,xn)dnx
E,! „ = /
(16)
J —3
E{x\,...,xn)
being the probability density of finding the n eigenvalues in A at
\X\, . . . , Xn).
Useful expressions for E (x\,. • •, xn) and En can be found in Refs. 2 and 3. It is expressible in terms of a determinant E{xi,...,xn)
=det Lp(xi\xj)
;
(i, j) e ( 1 . . .n)
(17)
where
L
'=^ k 153
(18)
2006
H.
Kunz
Kp is an operator whose kernel in the simplest case j3 = 2 is given by
K y) =
>w
(19)
^hf
defined on L2(—s, s). Universality comes from the fact that E(xi,...,x„) is expressible in terms of the correlation functions and the latter ones depends only on /3, for a large class of W. W modifies only the density of states and therefore the mean level spacing /(e). This expression for p (R\r) is mostly useful in the small window limit, because when s->0 En ~ s f « 2 +"d-§)
(20)
Moreover in this case we have lim snE (sxi ,...,sxn)-An
TT
s—>0
Jb
(21)
Jj
-*•-*•
\
so that the probability distribution of R shows a scaling behavior s'13-1pr
lim
•
1-R (TTTS)
s->0 i2—»• 1
>X 2 —
= f gp(\)d\
(22)
Jx
the function gp{\) being given by 5/3 (A) =
A0\—
^Vl-A+^lnA-lnl + vT^A
(23)
On the other hand, one can see from eq (13) and (14) that the probability distribution of R is well defined at infinite times. Namely (24)
pn(R\T)=Pn(R\oo)+Olas can be seen by an integration by parts where oo
/-27T "
I
^i
R-
/
Hn{zi,...,Zn)
I
Y[
6
n
£ **»'
(25)
3=1
~^f \
Using an integral representation for the delta appearing in the definition (10) of the jUn, we can reexpress (25) as pe+ioo /•e+joo
i
pn (R\oo) =
/
f+oo /-+oo
dueu j
r
drrJ0 h/Hr)
/.oo
/
dze~uzJQ (rz)
-3
-1
(26)
e being any positive number, and Jo(x) the Bessel function. This expression can be simplified, considerably when (3 = 2, 4. In the unitary case (/? = 2) one finds n — 1
pn(R\oo)
= —-{l-R)-*154
n-3
(27)
Quantum
Dynamics
and Random Matrix Theory
2007
For a large window of energy, it is more appreciate to find another expression for pn (R\T). It is given as some integral over a Fredholm determinant Q. Q is a generating function for the variables {yj} and {XJ} appearing in the definition of R, eq (2). I
\
N
Q (r; ip; z) = lim / exp —iN \]
yjXj [r cos (2TTTXJ +
Z}\
(28)
It can be expressed in terms of the operator Kp appearing in eq (19), when /3 = 1, 2 as 9 = E0 det (l +
K/sg^
(29)
Eo = [det (1 - Kp)]*
(30)
with
and g is the multiplication operator by the function 2i 1 + — [z + r COS (27TTX + if)]
(31)
When the window is large (s » 1) we can expand the determinant in powers of K/3, the first two terms of this expansion dominating the other ones.4 One finds that the probability distribution is exponential. ,. 1 f R lim —p s—>-oo 5
1 T
| = —T-r a{r) exp
R a(r)
(32)
In the orthogonal case (/3 = 1), for example 4 - 2 | r | + | r | l n l + 2|r| O-(T)
,2+Mln§£|±i
if |r| < 1
if|r|>l
(33)
One can notice the singularity at the Heisenberg time r — 1 and the fact that
fTl
1
/ n - T0 j T n then we get a self-averaging quantity lim -p (-)
ir R{r)di
(34)
=S(R-a)
(35)
dTo-(r)
(36)
with fTl
1 a =
I
n - r0 JTa
Some numerical work on chaotic billiards, 5 in the large window limit, confirm this exponential distribution. Integrable billiards show a very different behaviour.5 155
2008 H. Kunz Finally, we would like to mention the fact that Wigner's energy level statistics can be obtained for models, where eigenvalues and eigenvectors are correlated. We think therefore that the study of quantum dynamics could discriminate between such models and those we have considered where they are uncorrelated. Acknowledgements This work is dedicated to F.Y. Wu for his 70*'1 birthday. References 1. See for example the review by O. Bohigas, Les Houches 1989, Chaos et physique quantique (M-J. Giannoni et al, North Holland, 1991). 2. M.L. Mehta, Random Matrices (Ed. Academic Press Inc, 1991). 3. C.A. Tracy and H. Widom, Journal of Statistical Physics 92, 809 (1998) 4. H. Kunz, J. Phys. A : Math & Gen 32, 2171 (1999) 5. R. Aurich and F. Steiner, Int. J. Mod. Phys. 13, 2361 (1999)
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International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2009-2015 © World Scientific Publishing Company
I N T E G R A B L E COUPLING I N A M O D E L F O R J O S E P H S O N T U N N E L I N G BETWEEN N O N - I D E N T I C A L B C S S Y S T E M S
JON LINKS* Department
of Mathematics, The University Brisbane, 4072, Australia.
of
Queensland,
KATPJNA E. H I B B E R D t Departamento
de Fisica Teorica, Universidad Zaragoza, Spain.
de Zaragoza,
Received 14 November 2001 We extend a recent construction for an integrable model describing Josephson tunneling between identical BCS systems to the case where t h e BCS systems have different single particle energy levels. The exact solution of this generalized model is obtained through the Bethe ansatz.
1. Introduction. The experimental work of Ralph, Black and Tinkham 1 ' 2 on the discrete energy spectrum in small metallic aluminium grains has generated substantial interest in understanding the nature of superconducting correlations at the nano-scale level. Their results indicate significant parity effects due to the number of electrons in the system. For grains with an odd number of electrons, the gap in the energy spectrum reduces with the size of the system, in contrast to the case of a grain with an even number of electrons, where a gap larger than the single electron energy levels persists. In the latter case the gap can be closed by a strong applied magnetic field. The conclusion drawn from these results is that pairing interactions are prominent in these nano-scale systems. For a grain with an odd number of electrons there will always be at least one unpaired electron, so it is not necessary to break a Cooper pair in order to create an excited state. For a grain with an even number of electrons, all excited states have a least one broken Cooper pair, resulting in a gap in the spectrum. In the presence of a strongly applied magnetic field, it is energetically more favourable for a grain with an even number of electrons to have broken pairs, and hence in this case there are excitations which show no gap in the spectrum. * email: [email protected]. t email: [email protected] 157
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J. Links & K. E. Hibberd
The physical properties of a small metallic grain are described by the reduced BCS Hamiltonian 3 ' 4 c c HBCS
= Y^ eini ~ 9 Y, 4 + 4 - C J - C J + J=l
W
j,k
Above, j = 1,..., C labels a shell of doubly degenerate single particle energy levels with energies ej and rij is the fermion number operator for level j . The operators c j±i cl± a r e t r i e annihilation and creation operators for the fermions at level j . The labels ± refer to time reversed states. One of the prominent features of the Hamiltonian (1) is the blocking effect. For any unpaired electron at level j the action of the pairing interaction is zero since only paired electrons are scattered. This means that the Hilbert space can be decoupled into a product of paired and unpaired electron states in which the action of the Hamiltonian on the subspace for the unpaired electrons is automatically diagonal in the natural basis. In view of the blocking effect, it is convenient to introduce hard-core boson operators bj = Cj-Cj+, b', = c]+Cj_ which satisfy the relations
(6})2 = 0, [6 j ,6t] =5j . fc (i_26t 6 .) [bj,bk} =
[b],b{}=0
on the subspace excluding single particle states. In this setting the hard-core boson operators realise the su(2) algebra in the pseudo-spin representation, which will be utilized below. The original approach of Bardeen, Cooper and Schrieffer5 to describe the phenomenon of superconductivity was to employ a mean field theory using a variational wavefunction for the ground state which has an undetermined number of electrons. The expectation value for the number operator is then fixed by means of a chemical potential term fi. One of the predictions of the BCS theory is that the number of Cooper pairs in the ground state of the system is given by the ratio A/d where A is the BCS "bulk gap" and d is the mean level spacing for the single electron eigenstates. For nano-scale systems, this ratio is of the order of unity, in seeming contradiction with the experimental results discussed above. The explanation for this is that the mean-field approach is inappropriate for nano-scale systems due to large superconducting fluctuations. As an alternative to the BCS mean field approach, one can appeal to the exact solution of the Hamiltonian (1) derived by Richardson and Sherman. 6 It has also been shown by Cambiaggio, Rivas and Saraceno7 that (1) is integrable in the sense that there exists a set of mutually commutative operators which commute with the Hamiltonian. These features have recently been shown to be a consequence of the fact that the model can be derived in the context of the Quantum Inverse Scattering Method (QISM) using a solution of the Yang-Baxter equation associated with the Lie algebra su(2).8'9 One of the aims of the present work is to extend this approach 158
Integrable Coupling in a Model for Josephson Tunneling 2011
for application to generalised models. As a specific example, we will show that a model for strong Josephson coupling between two BCS systems falls into this class. Recall first that electron pairing interactions manifest themselves in macroscopic systems via three well known phenomena: • supercurrents • Meissner effect • Josephson effect As noted by von Delft,4 the notion of a supercurrent in a nano-scale system is inapplicable because the mean fee path of an electron is comparable to the system size. Likewise, the penetration depth of an applied magnetic field is comparable to the system size, which prohibits any Meissner effect. Josephson 11 put forth a proposal for the tunneling of electron pairs between superconductors separated by an insulating barrier. A theory was derived to describe weak coupling between two superconductors treated at the mean field level in the grand-canonical ensemble. A remarkable prediction of the theory was that it is possible for a direct current to flow across the insulator for the case of zero applied voltage, whereas a constant voltage across the insulator produces an alternating current. The essential features of the theory stem from the phase difference between the superconductors, which is well defined since the variational wavefunctions for the superconductors have undetermined particle numbers. For the case of nano-scale systems, the above predictions are again invalid due to the finite particle numbers for each system, giving rise to phase uncertainty. However, if we are to consider strong coupling where individual particle numbers are not conserved, only total particle number, it is appropriate to study the effective Hamiltonian C
H = HBCS(1) + HBCSP) -£JJ2
O'lWMS) + 6}(2)6fc(l)) ,
(2)
i,fc where ej is the Josephson coupling energy, for the purpose of investigating the nature of pair tunneling at the nano-scale level. In a previous work10 it was shown that the above Hamiltonian is integrable for ej = g for the case when HBCS(1), HBCS{1) have identical single electron energy levels. Below we will extend this construction to the case where HBCSQ), HBCS(2) describe non-identical systems.
2. A universal integrable system. First we introduce the Lie algebra su(2) with generators S+, S~, Sz satisfying the commutation relations [Sz, 5±] = ±S±,
[S+, S~] = 2SZ. 159
(3)
2012
J. Links & K. E. Hibberd
The Casimir invariant, which commutes with each element of the algebra, has the form C = S+S- + S~S+
+ 2 (Szf
.
Associated with the su(2) algebra there is a solution of the Yang-Baxter equation in EndF ® EndV ® su(2), where V denotes a two-dimensional vector space. This solution reads 12 Ru(u - v)Li(u)Li2(v)
= L2(v)Li(u)Ri2(u
- v)
with 2
u '-^ m,n
L(u) = I I + ^ (e\ ® Sz - e\ ® Sz + e\ ® 5 " + e 2 ® S+) where {e™} are 2 x 2 matrices with 1 in the (m,n) entry and zeroes elsewhere. Above, / is the identity operator and r/ is a scaling parameter for the rapidity variable u which plays an important role in the subsequent analysis. With this solution we construct the transfer matrix t(u) = tr 0 (G0Loc(u - ec)-L0i(u
- ei))
(4)
which is an element of the £-fold tensor algebra of su(2). Above, tro denotes the trace taken over the auxiliary space and G = exp(ar7a) with a = diag(l, —1). A consequence of the Yang-Baxter equation is that [t(u), t(v)] = 0 for all values of the parameters u and v, and independent of the representations of su(2) in the tensor algebra. Defining Tj = hm
z-Z-tiu)
for j — 1,2, ...,£, we may write in the quasi-classical limit Tj = Tj + 0(77) and it follows that [TJ, rfe] = 0, \/j, k. Explicitly, these operators read
rj=2aSz + f:-^W with 6 = S+ ® S' + S~ S+ + 2SZ (8) Sz. We define a Hamiltonian through £ c 3,k
3
a
€j
(5) €k
c j
A
c 3
i,k
The Hamiltonian is universally integrable since it is clear that [H, Tj] = 0, V? irrespective of the realizations of the su(2) algebra in the tensor algebra. 160
Integrable Coupling in a Model for Josephson Tunneling
2013
Realizing the su(2) generators through the hard-core bosons; viz s
t=b*>
s
i=bh
^=\{i-nj)
(8)
one obtains (1) (up to a constant) with g = 1/a as shown by Zhou et al. 8 and von Delft and Poghossian. 9 We now turn to applying (7) for the study of two coupled BCS systems. To accommodate this, it is convenient to first consider three index sets Po, Pi, Pi such that individually the BCS Hamiltonians are expressible c c
HBcs(i)=
e n
Yl
J J-9
Yl
j6(P0UPi)
b b
kr
j,k€{PoUPi)
If the single particle energy €j is common to both systems, then j G Po. Hence it is meant to be understood that ej ^ eft ^ q V j G Pi, k G P2, I G P3. In the case that j G Po, the local su(2) operators are described by the tensor product of two pseudo-spin realisations acting on the four-dimensional tensor product space. We can now realise (7) in terms of the hard-core boson representation (8) Sf = bj(i),
Sr=b](i),
5 ; = !(/-«,•(»))
for j G Pi, i — 1,2 whereas for j G Po we take the tensor product representation
Sf = Ml) + 6,(2) Sj=b){l)
+ b){2)
S;=/-5(n,-(l)+n,(2)). Under this representation of (7) we obtain (2) with ej — g = 1/a, establishing integrability at this value of the Josephson coupling energy. For the case when the index sets P i , P2 are both empty, i.e., the two BCS systems are identical, this result was previously shown by Links et al.. 10 3. T h e exact solution. In addition to proving integrability for ej = g, we can also obtain the exact solution from the Bethe ansatz. Below we will derive the energy eigenvalues for the Hamiltonian (7) in a very general context, which includes those of (2) with ej = g as a particular case. For each index k in the tensor algebra in which the transfer matrix acts, and accordingly in (7), suppose that we represent the su(2) algebra through the irreducible representation with spin Sfc. Thus {S~£, S^, S%} act on a (2sfc + l)-dimensional space. Employing the standard method of the algebraic Bethe ansatz 12 gives that the eigenvalues of the transfer matrix (4) take the form c , M J A(w) = exp(m7) I I I •L,-L k
U — 6k
X ±
3 161
U — Wj
2014
J. Links & K. E. Hibberd C
I
\ TT
M
u ~~ €efc V$k T T u -Wj + f] k ~- VSk
U
x x
U-tk K
U-
Wj
J
Above, the parameters Wj are required to satisfy the Bethe ansatz equations exp(2m?)
IT "»»-6fc+w V- wi - ek - r/Sfe
=
_ f r m-iPj+i^ -"-A tuj -Wj -r)
K
J
The eigenvalues of the conserved operators (5) are obtained through the appropriate terms in the expansion of the transfer matrix eigenvalues in the parameter X]. This yields the following result for the eigenvalues Xj of Tj
such that the parameters Vj satisfy the coupled algebraic equations 2Q +
V
-
^
= ; r ^ — .
(10)
Through (9) we can now determine the energy eigenvalues of (7). It is useful to note the following identities M M c VjSk i
M
• 3
t
k
C
v
v
i ~ ek
^
J
V 3
€k
l
t k ML
C
k
3
M
^-
£ f c
C V
k
J J
~ek
v k
Employing the above it is deduced that c c A 2a 2aM
E j = E ^ ' ~~ j
C
j C
M
M
e
C
s sfc 2 M
E A- = 2a E w + E E J 3
3
3
which, combined with the eigenvalues yields the energy eigenvalues
E k
k^3
%SJ(SJ
M
M
sfc
2a
E vi + M(M -x)
3
+ 1) for the Casimir invariants Cj,
C
£ = 2E«j-2E s * 6 *3
(n)
k
From the above expression we see that the quasi-particle excitation energies are given by twice the Bethe ansatz roots {VJ} of (10). 162
Integrable Coupling in a Model for Josephson Tunneling 2015 In order to specialise this result t o (2) a t integrable coupling, it is useful to first make the following observation. For j £ Po, in which case t h e su(2) algebra is realised via the tensor product of t w o hard-core boson representations, it is well known t h a t the representation space is completely reducible into triplet states and a singlet state. Note however, t h a t for t h e singlet s t a t e t h e su(2) generators act trivially, and hence this state is blocked from scattering in analogy with t h e blocking of single particle states discussed in t h e introduction. Hence t h e su{2) algebra will only act non-trivially on t h e triplet s t a t e s . In specialising (10,11) t o t h e case of (2), we need only to set Sj = 1/2 for j G Pi U Pi a n d Sj = 1 for j £ P0.
4. Conclusion We have displayed t h e existence of a general class of integrable systems which includes the reduced BCS H a m i l t o n i a n a n d a model for strong Josephson tunneling between two reduced BCS systems. B y deriving t h e models t h r o u g h t h e QISM we have also determined t h e exact solution via t h e B e t h e ansatz. A further application of this approach is t h e c o m p u t a t i o n of form factors a n d correlation functions. 8 ' 1 0
Acknowledgement s J o n Links thanks Departamento d e Fisica Teorica de Universidad de Zaragoza for hospitality and Prof. M.-L. Ge for t h e k i n d invitation to a t t e n d t h e A P C T P - N a n k a i Joint Symposium celebrating t h e 7 0 t h b i r t h d a y of Prof. F.Y. W u . We t h a n k the Australian Research Council a n d t h e Ministerio de E d u c a t i o n y Cultura, Espana for financial support.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
C.T. Black, D.C. Ralph and M. Tinkham, Phys. Rev. Lett. 76, 688 (1996). D.C. Ralph, C.T. Black and M. Tinkham, Phys. Rev. Lett. 78, 4087 (1997). J. von Delft and D. C. Ralph, Phys. Rep. 345, 61 (2001). J. von Delft, Ann. Phys. 10, 219 (2001). L.N. Cooper, J. Bardeen and J.R. Schrieffer, Phys. Rev. 108, 1175 (1957). R.W. Richardson, Phys. Lett. 3, 277 (1963); 5, 82 (1963); R.W. Richardson and N. Sherman, Nucl. Phys. 52, 221 (1964); 52, 253 (1964). M.C. Cambiaggio, A.M.F. Rivas and M. Saraceno, Nucl. Phys. A 6 2 4 , 157 (1997). H.-Q. Zhou, J. Links, R.H. McKenzie and M.D. Gould, cond-mat/0106390. J. von Delft and R. Poghossian, cond-mat/0106405. J. Links, H.-Q. Zhou, R.H. McKenzie and M.D. Gould, cond-mat/0110105. B.D. Josephson, Phys. Lett. 1, 251 (1962). L.D. Faddeev, Int. J. Mod. Phys. A 1 0 , 1845 (1995).
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International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2017-2032 © World Scientific Publishing Company
F I N I T E D E N S I T Y A L G O R I T H M I N LATTICE QCD—A CANONICAL ENSEMBLE APPROACH
KEH-FEI LIU* Department
of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA
Received 25 January 2002 Final Form 8 May 2002 I will review t h e finite density algorithm for lattice QCD based on finite chemical potential and summarize the associated difficulties. I will propose a canonical ensemble approach which projects out the finite baryon number sector from the fermion determinant. For this algorithm to work, it requires an efficient method for calculating the fermion determinant and a Monte Carlo algorithm which accommodates unbiased estimate of t h e probability. I shall report on the progress made along this direction with the Pade-Z2 estimator of the determinant and its implementation in the newly developed Noisy Monte Carlo algorithm.
1. Introduction Fermions at finite density or finite chemical potential is a subject of a wide range of interest. It is relevant to condensed matter physics, such as the Hubbard model away from half-filling. The research about nuclei and neutron stars at low and high nucleon density is actively pursued in nuclear physics and astrophysics. The subject of quark gluon plasma is important for understanding the early universe and is being sought for in relativistic heavy-ion collisions in the laboratories. Furthermore, speculation about color superconducting phase has been proposed recently for quantum chromodynamics (QCD) at very high quark density.1 Although there are models, e.g. chiral models and the Nambu-Jona-Lasinio model which have been used to study QCD at finite quark density, the only way to study QCD at finite density and temperature reliably and systematically is via lattice gauge calculations. There have been extensive lattice calculations of QCD at finite temperature. 2 On the contrary, the calculation at finite density is hampered by the lack of a viable algorithm. In this talk, I shall first review the difficulties associated with the finite density algorithm with chemical potentials in Sec. 2.1 will then outline in Sec. 3 a proposal for a finite density algorithm in the canonical ensemble which projects out the *email: [email protected] 165
2018
K.-F.
Liu
nonzero baryon number sector from the fermion determinant. In Sec. 4, a newly developed Noisy Monte Carlo algorithm which admits unbiased estimate of the probability is described. Its application to the fermion determinant is outlined in Sec. 5. I will discuss an efficient way, the Pade-Z2 method, to estimate the Tr log of the fermion matrix in Sec. 6. The recent progress on the implementation of the Kentucky Noisy Monte Carlo algorithm to dynamical fermions is presented in Sec. 7. Finally, a summary is given in Sec. 8. 2. Finite Chemical Potential The usual approach to the finite density in the Euclidean path-integral formalism of lattice QCD is to consider the grand canonical ensemble with the partition function ZGC{»)
= Y,
Z
Ne~'iN = fvu
N
det M[U, fi]e-s°M,
(i)
•*
where the fermion fields with fermion matrix M has been integrated to give the determinant. U is the gauge link variable and Sg is the gauge action. The chemical potential is introduced to the quark action with the e^a factor in the time-forward hopping term and e~Ma in the time-backward hopping term. Here a is the lattice spacing. However, this causes the fermion action to be non-Hermitian, i.e. 75M75 ^ M. As a result, the fermion determinant det M[U] is complex and this leads to the infamous sign problem. There are several approaches to avoid the sign problem: 2.1. Fugacity
Expansion
It was proposed by the Glasgow group 3 that the sign problem can be circumvented based on the expansion of the grand canonical partition function in powers of the fugacity variable e M / T , B=3V
ZGC(ji/T,T,V)
=
Y,
e^TBZB(T,V),
(2)
B=-3V
where ZB is the canonical partition function for the baryon sector with baryon number B. ZQC is calculated with reweighting of the fermion determinant ZGc{fl)
(3) ~ <detM[tf,0] >M= °' Since the reweighting is based on the gauge configuration with \i = 0, it avoids the sign problem. However, this does not work, except perhaps at small // or near the finite temperature phase transition. We will dwell on this later in Sec. 3. This is caused by the 'overlap problem'4 where the important samples of configurations in the /x = 0 simulation has exponentially small overlap with those relevant for the finite density. As a result, the onset of baryon begins at \i ~ mn/2 instead of the expected M J V / 3 which resembles the situation of the quenched approximation.
166
Finite Density
2.2. Imaginary
Chemical
Algorithm
in Lattice QCD
2019
Potential
In this approach, the chemical potential is taking an imaginary value fi — iv. The fermion determinant is real in this case and one can avoid the sign problem. 5-7 The partition function is Zacliv/T,
T,V) = Tr e~fl/Tei^/T,
(4)
which is periodic with respect to v with a period of 2TVT. Comparing with Eq. (2), one can in principle obtain canonical partition function ZB from the Fourier transform ZB(T, V) = ^fJo
dvZGC{iv/T,
T, V)e-^T.
(5)
In this approach, one needs to integrate over the whole range of v from 0 to 2'KT after one obtains the Monte Carlo configurations of ZociivjT, T, V) at different v. In practice, it is proposed to calculate the following ratio in the two-dimensional Hubbard model, 7 ZGC(tvo/T,T,V)
J
V
K
o;
detM(^0)
V
'
with a reference value VQ. Several patches each centered around a different reference point VQ are used to cover the range of v. This was successful for the two-dimensional Hubbard model with a 4 2 x 10 lattice up to B — 6 where the determinant was calculated exactly. While this works for a small lattice in the Hubbard model, it would not work for reasonably large lattices in QCD. This is because the direct calculation of the determinant is a V3 (or V2 for a sparse matrix) operation which is an impracticable task for the quark matrix which is typically of the dimension 10 6 x 10 6 . Any stochastic estimation of the determinant will inevitably introduce systematic error. Furthermore, this will also suffer from the 'overlap' problem discussed above. Any Monte Carlo simulation at a reference point u0 will have exponentially small overlap with those configurations important to a nonzero baryon density. 2.3. Overlap Ensuring
Multi-parameter
Reweighting
To alleviate the sign problem with the real chemical potential and the overlap problem due to reweighting, it is proposed 8 to do the reweighting in the multiple parameter space. The generic partition function ZQC in Eq. (1) is parametrized by a set of parameters a, such as the chemical potential ^, the gauge coupling /3, the quark mass mq, etc. The partition function can be written to facilitate reweighting ZGC{a)=
[ VUdetM[U,Qo]e-^°o]{e-sg[E/,a]+s9[I/.«o] J
det
fJU a] detM[c7, aoj
where the Monte Carlo simulation is carried out with the ao set of parameters and the terms in the curly bracket are treated as observables. This is applied to study the end point in the T-/x phase diagram. In this case, the Monte Carlo simulation 167
2020
K.-F.
Liu
is carried out where the parameters in ao include fi — 0 and (3C which corresponds to the phase transition at temperature Tc. The parameter set a in the reweighted measure include mu ^ 0 and an adjusted /3 in the gauge action. The new (3 is determined from the Lee-Yang zeros so that one is following the transition line in the T-/i plane and the large change in the determinant ratio in the reweighting is compensated by the change in the gauge action to ensure reasonable overlap. This is shown to work to locate the transition line from fx = 0 and T = TC down to the critical point on the 4 4 and 6 3 x 4 lattices with staggered fermions.8 While the multi-parameter reweighting is successful near the transition line, it is not clear how to extend it beyond this region, particularly the T = 0 case where one wants to keep the /3 and quark mass fixed while changing the fi. One still expects to face the overlap problem in the latter case. For large volumes, calculating the determinant ratio will be subjected to the same practical difficulty as discussed in the previous section 2.2. 3. Finite Baryon Density — A Canonical Ensemble Approach We would like to propose an algorithm to overcome the overlap problem at zero temperature which is based on the canonical ensemble approach. To avoid the overlap problem, one needs to lock in a definite nonzero baryon sector so that the exponentially large contamination from the zero-baryon sector is excluded. To see this, we first note that the fermion determinant is a superposition of multiple quark loops of all sizes and shapes. This can be easily seen from the property of the determinant
detM = e ^
= l+ £ ^ ^ .
(8)
n=l
Upon a hopping expansion of log M, Tr log M represents a sum of single loops with all sizes and shapes. The determinant is then the sum of all multiple loops. The fermion loops can be separated into two classes. One is those which do not go across the time boundary and represent virtual quark-antiquark pairs; the other includes those which wraps around the time boundary which represent external quarks and antiquarks. The configuration with a baryon number one which entails three quark loops wrapping around the time boundary will have an energy MB higher than that with zero baryon number. Thus, it is weighted with the probability e~MBNtat compared with the one with no net baryons. We see from the above discussion that the fermion determinant contains a superposition of sectors of all baryon numbers, positive, negative and zero. At zero temperature where MsNtO-t > 1, the zero baryon sector dominates and all the other baryon sectors are exponentially suppressed. It is obvious that to avoid the overlap problem, one needs to select a definite nonzero baryon number sector and stay in it throughout the Markov chain of updating configurations. To select a particular baryon sector from the determinant can be achieved by the following procedure: 9 first, assign an U(l) phase factor e"1^ to the links between the time slices t and t + 1 so that the link U/W is multiplied 168
Finite Density Algorithm in Lattice QCD 2021
by e l<^/e**; then the particle number projection can be carried out through the Fourier transformation of the fermion determinant like in the BCS theory i r2* PN = TT #e-^NdetM[0] tit Jo
(9)
where N is the net particle number, i.e. particle minus antiparticle. Note that all the virtual quark loops which do not reach the time boundary will have a net phase factor of unity; only those with a net N quark loops across the time boundary will have a phase factor el(^N which can contribute to the integral in Eq. (9). Since QCD in canonical formulation does not break Z(3) symmetry, it is essential to take care that the ensemble is canonical with respect to triality. To this end, we shall consider the triality projection 9 ' 10 to the zero triality sector detM=^
]T
detM[0 +
fc27r/3].
(10)
k=0;±l
This amounts to limiting the quark number N to a multiple of 3. Thus the triality zero sector corresponds to baryon sectors with integral baryon numbers. Another essential ingredient to circumvent the overlap problem is to stay in the chosen nonzero baryon sector so as to avoid mixing with the zero baryon sector with exponentially large weight. This can be achieved by preforming the baryon number projection as described above before the accept/reject step in the Monte Carlo updating of the gauge configuration. If this is not done, the accepted gauge configuration will be biased toward the zero baryon sector and it is very difficult to project out the nonzero baryon sector afterwards. This is analogous to the situation in the nuclear many-body theory where it is known13 that the variation after projection (Zeh-Rouhaninejad-Yoccoz method 1 4 ' 1 5 ) is superior than the variation before projection (Peierls-Yoccoz method 16 ). The former gives the correct nuclear mass in the case of translation and yields much improved wave functions in mildly deformed nuclei than the latter. To illustrate the overlap problem, we plot in Fig.l Tr log M[(f>] for a configuration of the 8 3 x 12 lattice with the Wilson action with /3 = 6.0 and K = 0.150 which is obtained with 500 Z-z noises. We see that the it is rather flat in indicating that the Fourier transform in Eq. (9) will mainly favor the zero baryon sector. On the other hand, at finite temperature, it is relatively easier for the quarks to be excited so that the zero baryon sector does not necessarily dominate other baryon sectors. Another way of seeing this is that the relative weighting factor e-MBNtat can be 0(1) at finite temperature. Thus, it should be easier to project out the nonzero baryon sector from the determinant. We plot in Fig. 2 a similarly obtained Tr log M[<j>] for a configuration of the 8 x 20 2 x 4 lattice with /? = 4.9 and « = 0.182. We see from the figure that there is quite a bit of wiggling in this case as compared to that in Fig. 1 indicating that it is easier to project out a nonzero baryon sector through the Fourier transform at finite temperature. 169
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K.-F. Liu
83 x 12 /3 = 6.0 K = 0.150 Wilson action lattice ~i
1
1
1
r
854
852
850
$
f
f
^
^
^
l
>
^
^
-
E-.
846 j
i
i
i
i
0
2
4
6
8
(j) = 27T *
Fig. 1.
I
x/9
Trlog M[(j>] for a 8 3 x 12 configuration with Wilson action as a function of <j>.
We should mention that while we think we can overcome the overlap problem and the determinant detM[] is real in this approach, nevertheless in view of the fact that the Fourier transform in Eq. (9) involves the quark number N the canonical approach may still have the sign problem at the thermodynamic limit when N and V are very large. However, we think it might work for small N such as 3 or 6 for one or two baryons in a finite V. This should be a reasonable start for practical purposes. While it is clear what the algorithm in the canonical approach entails, there are additional practical requirements for the algorithm to work. These include an unbiased estimation of the huge determinant in lattice QCD and, moreover, a Monte Carlo algorithm which accommodates the unbiased estimate of the probability. We shall discuss them in the following sections.
4. A Noisy Monte Carlo A l g o r i t h m There are problems in physics which involve extensive quantities such as the fermion determinant which require V3 steps to compute exactly. Problems of this kind with large volumes are not numerically applicable with the usual Monte Carlo algorithm which require an exact evaluation of the probability ratios in the accept/reject step. 170
Finite Density Algorithm in Lattice QCD
82 1250
X
20 x 4 /? = 4.9
i
K =
2023
0.182 Wilson action lattice i
i
i
i
1245
P 1240 1235
-
$ -
$> $
gl230 ^
1225
£
1220
-
1215
-
1210
-
-
2> -
$
1205 lonn
l
i
1
1
1
0
2
4
6
8
(j> = 2n* Fig. 2.
x/9
Tr log M[<j>\ for a 8 x 20 2 x 4 finite temperature configuration with dynamical fermion.
To address this problem, Kennedy and Kuti 11 proposed a Monte Carlo algorithm which admits stochastically estimated transition probabilities as long as they are unbiased. But there is a drawback. The probability could lie outside the interval between 0 and 1 since it is estimated stochastically. This probability bound violation will destroy detailed balance and lead to systematic bias. To control the probability violation with a large noise ensemble can be costly. We propose a noisy Monte Carlo algorithm which avoids this difficulty with two Metropolis accept/reject steps. Let us consider a model with Hamiltonian H(U) where U collectively denotes the dynamical variables of the system. The major ingredient of the new approach is to transform the noise for the stochastic estimator into stochastic variables. The partition function of the model can be written as
J[DU]e-H^ J[DU][Dt]Ps(Of(U,t).
(11)
where f(U, £) is an unbiased estimator of e~H^ from the stochastic variable £ and P$ is the probability distribution for £. The next step is to address the lower probability-bound violation. One first 171
2024
K.-F. Liu
notes that one can write the expectation value of the observable O as (O) = j{DU}[DH\ Ps(0O(U) sign(f) \f(U,0\/Z,
(12)
where sign(f) is the sign of the function / . After redefining the partition function to be
Z = J[DU][Dt]Pt(t)\f(U,0\,
(13)
which is semi-positive definite, the expectation of O in Eq. (12) can be rewritten as (O) = (0(U) sign(/))/(sign(/)).
(14)
As we see, the sign of f(U, £) is not a part of the probability any more but a part in the observable. Notice that this reinterpretation is possible because the sign of /(£/, £) is a state function which depends on the configuration of U and £. It is clear then, to avoid the problem of lower probability-bound violation, the accept/reject criterion has to be factorizable into a ratio of the new and old probabilities so that the sign of the estimated f(U, £) can be absorbed into the observable. This leads us to the Metropolis accept/reject criterion which incidentally cures the problem of upper probability-bound violation at the same time. It turns out two accept/reject steps are needed in general. The first one is to propose updating of U via some procedure while keeping the stochastic variables £ fixed. The acceptance probability Pa is Pa(UuZ^U2,t)
= min(l,ffi^|{) .
(15)
The second accept/reject step involves the refreshing of the stochastic variables £ according to the probability distribution P^(0 while keeping U fixed. The acceptance probability is Pa(U,Ci -> U,&) = min(l, j - ^ j j ) .
(16)
It is obvious that there is neither lower nor upper probability-bound violation in either of these two Metropolis accept/reject steps. Furthermore, it involves the ratios of separate state functions so that the sign of the stochastically estimated probability f(U,£) can be absorbed into the observable as in Eq. (14). Detailed balance can be proven to be satisfied and it is unbiased. 12 Therefore, this is an exact algorithm. 5. Noisy Monte Carlo with Fermion Determinant One immediate application of NMC is lattice QCD with dynamical fermions. The action is composed of two parts - the pure gauge action Sg(U) and a fermion action Sp{U) = — TrlnM({7). Both are functionals of the gauge link variables U. 172
Finite Density Algorithm in Lattice QCD 2025
To find out the explicit form of f(U,£), we note t h a t t h e fermion determinant can be calculated stochastically as a r a n d o m walk process 1 7 TrinM
, ™, ,,/TrlnM,„ T r l n M . ... j + Tj.^ ^ _| (1 + (...))) . ^ o This can be expressed in the following integral e TrlnM =
.„„, ( 17 )
e TrlnM
[1 + lyf l n M r ^ l + 0(p2 - ^)r,\ lnMr? 2 (l + 0(p 3 - \)4
lnMV3(...},
(18)
where P^ffe) is the probability distribution for the stochastic variable r\i. It can be the Gaussian noise or the Z 2 noise [Pn{j]i) — S(\r]i\ — 1) in this case). The latter is preferred since it has the minimum variance. 18 pn is a stochastic variable with uniform distribution between 0 and 1. This sequence terminates stochastically in finite time and only the seeds from the pseudo-random number generator need to be stored in practice. The function f(U,r],p) ( £ in Eq. (11) is represented by two stochastic variables rj and p here) is represented by the part of the integrand between the the square brackets in Eq. (18). One can then use the efficient Pade-Z 2 algorithm 19 to calculate the 77, lnM^i in Eq. (18). We shall discuss this in the next section. Finally, there is a practical concern that T r l n M can be large so that it takes a large statistics to have a reliable estimate of e ^ l n M from the series expansion in Eq. (18). In general, for the Taylor expansion ex = J^a; n /n!, the series will start to converge when xn/n\ > xn+1/(n + l)\. This happens at n = x. For the case x = 100, this implies that one needs to have more than 100! stochastic configurations in the Monte Carlo integration in Eq. (18) in order to have a convergent estimate. Even then, the error bar will be very large. To avoid this difficulty, one can implement the following strategy. First, one notes that since the Metropolis accept/reject involves the ratio of exponentials, one can subtract a universal number XQ from the exponent x in the Taylor expansion without affecting the ratio. Second, one can use a specific form of the exponential to diminish the value of the exponent. In other words, one can replace ex with (e(^-^o)/N^JV t o s a tj s fy \x _ x0\/N < 1. The best choice for xo is x, the mean of x. In this case, the variance of ex becomes es /N — 1. 6. The Pade-Z2 Method of E s t i m a t i n g Determinants Now we shall discuss a very efficient way of estimating the fermion determinant stochastically. 19 6.1. Pade
approximation
The starting point for the method is the Pade approximation of the logarithm function. The Pade approximant to log(z) of order [K, K] at ZQ is a rational function 173
2026
K.-F.
Liu
N(z)/D(z) where deg N(z) — deg D(z) = K, whose value and first 2K derivatives agree with logz at the specified point ZQ. When the Pade approximant N(z)/D(z) is expressed in partial fractions, we obtain
whence it follows K
log det M = Tr logM « 60TrI + £
bk • Tr(M + c f c I) _ 1 .
(20)
fc=i
The Pade approximation is not limited to the real axis. As long as the function is in the analytic domain, i. e. away from the cut of the log, say along the negative real axis, the Pade approximation can be made arbitrarily accurate by going to a higher order [K, K] and a judicious expansion point to cover the eigenvalue domain of the problem. 6.2. Complex
Z? noise trace
estimation
Exact computation of the trace inverse for N x N matrices is very time consuming for matrices of size N ~ 10 6 . However, the complex Z2 noise method has been shown to provide an efficient stochastic estimation of the trace. 1 8 ' 2 0 ' 2 1 In fact, it has been proved to be an optimal choice for the noise, producing a minimum variance.22 The complex Z2 noise estimator can be briefly described as follows.18'22 We construct L noise vectors r?1,7/2, • • •, rjL where if = {i^, rf2, rf3, • • •, rfN}T, as follows. Each element rfn takes one of the four values {±1, ±1} chosen independently with equal probability. It follows from the statistics of rfn that 1 E[<
Vn
>} = E[- £
L
1 L E[< V*mVn >] = E[- £
rfn] = 0,
r « ] = <W
The vectors can be used to construct an unbiased estimator for the trace inverse of a given matrix M as follows: L
N
E[< r/tM-S >] E £ [ ^
^
1
rtiM-)njn]
= E AC + ( £ M-]n)[± £ « ] =
TrM"1.
The variance of the estimator is shown to be 22 a2M = Var[< ^ M " 1 ^ >] = E [| < ^M" 1 ?? > - T r M" 1 ] 2 ] 1
N
1
N
= - £ M-\(M-\r = z £ IM-^I2 . 174
(21)
Finite Density
Algorithm in Lattice QCD
2027
The stochastic error of the complex Z2 noise estimate results only from the off-diagonal entries of the inverse matrix (the same is true for Z n noise for any n). However, other noises (such as Gaussian) have additional errors arising from diagonal entries. This is why the Z2 noise has minimum variance. For example, it has been demonstrated on a 16 3 x 24 lattice with j3 — 6.0 and K = 0.148 for the Wilson action that the Z2 noise standard deviation is smaller than that of the Gaussian noise by a factor of 1.54.18 Applying the complex Z2 estimator to the expression for the TrlogM in Eq. (20), we find
5> fc Tr(M + Cfe)'1 k
= iEEwt^,
(22)
where £*••>' = (M + cfcI)~ V are the solutions of (M + ckI)dk'j = r?,
(23)
Since M+Cf.1 are shifted matrices with constant diagonal matrix elements, Eq. (23) can be solved collectively for all values of Ck within one iterative process by several algorithms, including the Quasi-Minimum Residual (QMR), 23 Multiple-Mass Minimum Residual (M 3 R), 2 4 and GMRES. 25 We have adopted the M 3 R algorithm, which has been shown to be about 2 times faster than the conjugate gradient algorithm, and the overhead for the multiple Cfc is only 8%.26 The only price to pay is memory: for each c^, a vector of the solution needs to be stored. Furthermore, one observes that Cfc > 0. This improves the conditioning of (M + c^I) since the eigenvalues of M have positive real parts. Hence, we expect faster convergence for column inversions for Eq. (23). In the next section, we describe a method which significantly reduces the stochastic error. 6.3. Improved
PZ estimation
with unbiased
subtraction
In order to reduce the variance of the estimate, we introduce a suitably chosen set of traceless N x N matrices Q^ p \ i.e. which satisfy 5Z n = 1 Qn,n = 0, p = 1 • • • P. The expected value and variance for the modified estimator < ry^(M_1 — J2p=i ^ P Q ( P ) ) » ? > are given by p
E[< ^ ( M
-1
- Y^ APQ(p)))r/ >] = Tr 1VT1 , P=i
175
(24)
2028
K.-F. Liu
A M ( A ) = Var[< r ^ M "
1
- £
A p Q ^ f a >}
P=i
(25) p=i
771^71
for any values of the real parameters Xp. In other words, introducing the matrices Q( p ) into t h e estimator produces no bias, b u t m a y reduce the error bars if t h e Q( p ) are chosen judiciously. Further, Xp may b e varied at will to achieve a m i n i m u m variance estimate: this corresponds to a least-squares fit to the function rj^'M.~1r) sampled a t points rjj, j = 1---L, using t h e fitting functions { l ^ ^ Q ^ r / } , p = I--P. We now t u r n to the question of choosing suitable traceless matrices Q( p ) to use in t h e modified estimator. One possibility for t h e Wilson fermion matrix M = I — KD is suggested by the hopping parameter — K expansion of the inverse matrix, ( M + Cfcl)-1^
M + c,I
1 + Cfc
(l
+
Cfe
(1 + Cfc)
)(l_IT^-ID)
rD +
( l + cfe):
D2 +
(1 + cfe)
rD3
.(26)
This suggests choosing the matrices Q( p ) from among those matrices in the h o p p i n g p a r a m e t e r expansion which are traceless:
Q(1 Q<2 Q<3-
Q«
rD, (1 + Cfc) _2 K
(1+Cfc) (1+Cfc)
:&
rD3,
ft4
(D4 - TrD4),
(1 + ckf D5 (l + cfc)6 (1 + Cfc
7
(Db-TrDb),
Q(2r+1 (1 + Cfc \2r+2
D 2r+l
r = 3,4,5,-
It m a y be verified that all of these matrices are traceless. In principle, one c a n include all t h e even powers which entails t h e explicit calculation of all the allowed loops in TrD2r. In this manuscript we have only included Q ^ 4 \ Q ^ 6 \ and Q ( 2 r + 1 ) . 176
Finite Density
6.4. Computation
of
Algorithm
in Lattice QCD
2029
TrlogM
Our numerical computations were carried out with the Wilson action on the 8 3 x 12 (N = 73728) lattice with (5 = 5.6. We use the HMC with pseudofermions to generate gauge configurations. With a cold start, we obtain the fermion matrix M i after the plaquette becomes stable. The trajectories are traced with r = 0.01 and 30 molecular dynamics steps using K = 0.150. M2 is then obtained from M i by an accepted trajectory run. Hence M i and M2 differ by a continuum perturbation, and log[detMi/detM 2 ] ~ 0(1). We first calculate log detMi with different orders of Pade expansion around ZQ = 0 . 1 and ZQ — 1.0. We see from Table 1 that the 5th order Pade does not give the same answer for two different expansion points, suggesting that its accuracy is not sufficient for the range of eigenvalues of M i . Whereas, the 11th order Pade gives the same answer within errors. Thus, we shall choose P [ l l , l l ] ( z ) with ZQ = 0.1 to perform the calculations from this point on. Table 1. Unimproved and improved PZ estimates for log [detMi] with 100 complex Z2 noise vectors, K = 0.150.
P[K,K](z) z0 = 0.1 20 = 1.0
K = Original: Improved: Original: Improved:
5 473(10) 487.25(62) 798(10) 812.60(62)
7 774(10) 788.17(62) 798(10) 812.37(62)
9 796(10) 810.83(62) 798(10) 812.36(62)
11 798(10) 812.33(62) 799(10) 812.37(62)
In Table 2, we give the results of improved estimations for T r l o g M i . We see that the variational technique described above can reduce the data fluctuations by more than an order of magnitude. For example, the unimproved error 60 = 5.54 in Table 2 for 400 Z2 noises is reduced to 6n = 0.15 for the subtraction which includes up to the Q 1 1 matrix. This is 37 times smaller. Comparing the central values in the last row (i.e. the l l t / l order improved) with that of unimproved estimate with 10,000 Z2 noises, we see that they are the same within errors. This verifies that the variational subtraction scheme that we employed does not introduce biased errors. The improved estimates of Tr log M i from 50 Z2 noises with errors Sr from Table 2 are plotted in comparison with the central value of the unimproved estimate from 10,000 noises in Fig. 3. 7. Implementation of the Kentucky Noisy M o n t e Carlo Algorithm We have recently implemented the above Noisy Monte Carlo algorithm to the simulation of lattice QCD with dynamical fermions by incorporating the full determinant directly.28 Our algorithm uses pure gauge field updating with a shifted gauge 177
2030
K.-F.
Liu
Table 2. Central values for improved stochastic estimation of log[det M i ] and rth—order improved Jackknife errors 5 r are given for different numbers of Z2 noise vectors, re is 0.150 in this case.
#z 2 0th So
1 St
Si 2na 52 3rd
53 4** 54 5th 55 6ts
56 •jts
Sr gth,
59 lltft 5n
50 802.98 ±14.0 807.89 ±4.65 813.03 ±2.46 812.07 ±1.88 812.28 ±1.20 813.50 ±0.82 813.54 ±0.67 814.18 ±0.44 813.77 ±0.40 813.54 ±0.38
100 797.98 ±9.81 811.21 ±3.28 812.50 ±1.65 812.01 ±1.31 812.52 ±0.94 813.07 ±0.62 813.23 ±0.49 813.74 ±0.36 813.62 ±0.30 813.41 ±0.27
200 810.97 ±7.95 814.13 ±2.48 811.99 ±1.34 811.79 ±1.01 812.57 ±0.68 813.36 ±0.47 813.22 ±0.35 813.44 ±0.26 813.49 ±0.22 813.45 ±0.21
400 816.78 ±5.54 815.11 ±1.84 812.86 ±1.01 812.44 ±0.74 812.86 ±0.48 813.70 ±0.34 813.28 ±0.25 813.42 ±0.19 813.40 ±0.16 813.44 ±0.15
600 815.89 ±4.47 814.01 ±1.50 811.87 ±0.83 812.18 ±0.58 812.85 ±0.39 813.47 ±0.29 813.20 ±0.21 813.39 ±0.16 813.43 ±0.14 813.43 ±0.13
800 813.10 ±3.83 814.62 ±1.29 812.89 ±0.72 812.99 ±0.51 813.25 ±0.35 813.54 ±0.25 813.37 ±0.18
1000 816.53 ±3.41 814.97 ±1.12 813.04 ±0.64 813.03 ±0.44 813.40 ±0.30 813.50 ±0.22 813.26 ±0.16
7
7
-
-
3000 813.15 ±1.97
10000 812.81 ±1.08
__
7 7 -
-
coupling to minimize fluctuations in the trace log is the Wilson Dirac matrix. It gives the correct results as compared to the standard Hybrid Monte Carlo simulation. However, the present simulation has a low acceptance rate due to the pure gauge update and results in long autocorrelations. We are in the process of working out an alternative updating scheme with molecular dynamics trajectory to include the feedback of the determinantal effects on the gauge field which should be more efficient than the pure gauge update. 8. S u m m a r y After reviewing the finite density algorithm for QCD with the chemical potential, we propose a canonical approach by projecting out the definite baryon number sector from the fermion determinant and stay in the sector throughout the Monte Carlo updating. This should circumvent the overlap problem. In order to make the algorithm practical, one needs an efficient way to estimate the huge fermion determinant and a Monte Carlo algorithm which admits unbiased estimates of the probability without upper unitarity bound violations. These are achieved with the Pade-Z2 estimate of the determinant and the Noisy Monte Carlo algorithm. So far, we have implemented the Kentucky Noisy Monte Carlo algorithm to incorporate dynamical fermions in QCD on a relatively small lattice and medium heavy quark 178
Finite Density Algorithm in Lattice QCD 2031
825
Improved estimates for Tr log M_1 I
1
1
-1
"
1
Improved at kappa=.150 ^—• Unimproved: 812.8 +/-1.1
820
815
—-J—-f—i
i
j
•
>
810 <
•
805 <>
800
1
4
1
1
6 8 order of subtraction
1
1
10
12
Fig. 3. The improved PZ estimate of TrlogMi with 50 noises as a function of the order of subtraction and compared to that of unimproved estimate with 10,000 noises. The dashed lines are drawn with a distance of 1 a away from the central value of the unimproved estimate. based on p u r e gauge updating. As a next step, we will work on a more efficient u p d a t i n g algorithm and project out the baryon sector to see if the finite density algorithm proposed here will live up to its promise. Acknowledgments This work is partially supported by the U.S. D O E grant DE-FG05-84ER40154. T h e a u t h o r wishes to thank M. Faber for introducing the subject of the finite density to him. Fruitful discussions with M. Alford, U.-J. Wiese, R. Sugar, and F . Wilczek are acknowledged. He would also thank the organizer, Ge Mo-lin for the invitation to a t t e n d t h e conference and his hospitality. References l. M. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett. B422, 247 (1998); R. Rapp, T. Schaefer, E.V. Shuryak, and M. Velkovsky, Phys. Rev. Lett. 8 1 , 53 (1998); for a review, see for example, K. Rajagopal and F. Wilczek, hep-ph/0011333. For recent review, see S. Ejiri, Nucl. Phys. B (Proc. Suppl.) 94, 19 (2001). I. M. Barbour, S. E. Morrison, E. G. Klepfish, J. B. Kogut, and M.-P. Lombardo, Nucl. Phys. (Proc. Suppl.) 60A, 220 (1998); I.M. Barbour, C.T.H. Davies, and Z. Sabeur, Phys. Lett. B215, 567 (1988). 179
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4. M. Alford, Nucl. Phys. B (Proc. Suppl.) 73, 161 (1999). 5. E. Dagotto, A. Moreo, R. Sugar, and D. Toussaint, Phys. Rev. B 4 1 , 811 (1990). 6. N. Weiss, Phys. Rev.D35, 2495 (1987); A. Hasenfratz and D. Toussaint, Nucl. Phys. B 3 7 1 , 539 (1992). 7. M. Alford, A. Kapustin, and F. Wilczek, Phys. Rev. D 5 9 , 054502 (2000). 8. Z. Fodor and S.D. Katz, hep-lat/0104001; J H E P 0203, 014 (2002) [hep-lat/0106002]. 9. M. Faber, Nucl. Phys. B444, 563 (1995) and private communication. 10. M. Faber, O. Borisenko, S. Mashkevich, and G. Zinovjev, Nucl. Phys. B(Proc. Suppl.) 42, 484 (1995). 11. A.D. Kennedy, J. Kuti, Phys. Rev. Lett. 5 4 (1985) 2473. 12. L. Lin, K. F. Liu, and J. Sloan, Phys. Rev. D 6 1 , 074505 (2000), [hep-lat/9905033]. 13. For reviews on the subject see for example C.W. Wong, Phys. Rep. 15C, 285 (1975); D.J. Rowe, Nuclear Collective Motion, Methuen (1970); P. Ring and P. Schuck, The Nuclear Many-Body Problem, Springer-Verlag (1980). 14. H.D. Zeh, Z. Phys. 188, 361 (1965). 15. H. Rouhaninejad and J. Yoccoz, Nucl. Phys. 78, 353 (1966). 16. R.E. Peierls and J. Yoccoz, Proc. Phys. Soc. (London) A 7 0 , 381 (1957). 17. G. Bhanot, A. D. Kennedy, Phys. Lett. 1 5 7 B , 70 (1985). 18. S. J. Dong and K. F. Liu, Phys. Lett. B 328, 130 (1994). 19. C. Thron, S. J. Dong, K. F. Liu, H. P. Ying, Phys. Rev. D 5 7 , 1642 (1998). 20. S.J. Dong, J.-F. Lagae, and K.F. Liu, Phys. Rev. Lett. 75, 2096 (1995); S.J. Dong, J.-F. Lagae, and K.F. Liu, Phys. Rev. D 5 4 , 5496 (1996). 21. N. Eicker, et al, SESAM-Collaboration, Phys. Lett. B389, 720 (1996). 22. S. Bernardson, P. McCarty and C. Thron, Comp. Phys. Commun. 78, 256 (1994). 23. A. Frommer, B. Nockel, S. Giisken, T h . Lippert and K. Schilling, Int. J. Mod. Phys. C 6 , 627 (1995). 24. U. Glassner, S. Giisken, Th. Lippert, G. Ritzenhofer, K. Schilling, and A. Frommer, hep-lat/9605008. 25. A. Frommer and U. Glassner, Wuppertal preprint BUGHW-SC96/8, to appear in SIAM J. Scientific Computing. 26. He-Ping Ying, S.J. Dong and K.F. Liu, Nucl. Phys. B(proc. Suppl.) 53, 993 (1997). 27. S. Duane, A. D. Kennedy, B. J. Pendleton, D. Roweth, Phys. Lett. B 1 9 5 (1987) 216. 28. B. Joo, I. Horvath, and K.F. Liu, hep-lat/0112033.
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International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2033-2038 © World Scientific Publishing Company
S H O R T - T I M E B E H A V I O R S OF L O N G - R A N G E D INTERACTIONS*
H. FANG, C. S. HE, Z. B. LI#, S. P. SETO Zhongshan
University,
Guangzhou
University,
Guangzhou,
China
Y. CHEN Guangzhou,
China
M. Y. WU Dongguan Institute
of Technology, Dongguan,
China
Received 15 January 2002
Recent results on the short-time behaviors of a few models possessing a common feature of long-ranged interaction will be summarized. For the disorder initial state, the initial order increase is observed for each model in a heat-bath at the critical temperature. The dynamic exponents are calculated. For arbitrary initial order and environment temperature, universal characteristic functions are introduced in order to generalize the scaling relations. Remarkable consistence between the theoretic renormalization group results and t h e simulations are found in the long-range regime.
The short-time phenomena are those which happen at the times just after a microscopic time-scale ^mic needed for a system to forget its microscopic details, and much smaller than the macroscopic time scale i m a c ~ r~vz. In this time regime, the system still remembers the macroscopic feature of the initial state. Since the pioneer work of H.K. Janssen et al., 1 universal short-time scaling has been found in a variety of different models (for a review, see Ref. 2). For initial states of zero correlation length and zero (or very small) initial order, the order increases in the short-time regime with a power law te where 6' is a characteristic exponent of the short-time regime. Recently, we generalized the results of Ref. 1 to systems with long-ranged interactions, 3 with an anisotropic cubic term, 4 ' 5 and with impurities. 6 ' 7 Hopefully these models could describe some realistic systems. On the other hand, we expect that the analytical calculations for long-ranged interactions based on expansions around the upper critical dimensions are more reliable in physical dimensions since *The research is supported by the National Natural Science Foundation of China and the Advance Research Center of Zhongshan University. * Corresponding author, E-mail: [email protected] 181
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the longer range of interaction the lower critical dimension(e = la — d with a the decay power of the interaction). A common observation is that the critical exponent 6' depends on the range of interactions. In the theoretic renormalization group calculation, the crossover from the long-ranged interaction(LRI) to the short-ranged interaction(SRI) is subtle and needs extra effort8 since the fixed point of LRI does not continuously approach that of the SRI as the interaction range decreases. There is a competition between two fixed points in the regime of the weakly long-ranged interaction. The exactly solvable kinetic spherical model provides a concrete example for the short-time behavior of LRI. 9 In one dimension, Monte Carlo simulation is possible. The theoretic results of LRI can be checked numerically. In order to have a picture for the crossover of LRI and SRI, we simulate an adsorption-desorption model which has a dynamic phase transition even for SRI. 1. Kinetic spherical model The Hamiltonian of the spherical model is
i
ij
with the constraint
£ S ? = JV
(2)
i
where i, j are labels of lattice sites, N is the total number of spins. In the dynamic process, a is a time-dependent Lagrange multiplier corresponding to the constraint. Joyce 10 first studied the static spherical model with long-ranged ferromagnetic interactions. In a d-dimensional lattice,
^ = W d + s ) /£^ ( d + s ) 3
with 0 < s < 2 for long-ranged interactions and s = a, while s > 2 for short-ranged interactions and a = 2. Where r^ is the distance between the sites i and j . The Langevin equation for this constrained spin system is dSj,
dt
-\aSi + \pJ2JvSJ+rIi
(3)
where A is the kinetic coefficient and rji being Gaussian white noises. A remarkable observation is that the ordering process (governed by the zerotemperature fixed point) and the critical dynamics (governed by the critical fixed point) can be uniquely described by a characteristic function. The universal characteristic function for arbitrary initial order 11 is also found analytically. The generalized scaling relation for the relative order mr = (m(t) — m(oo))/m(oo) are mr(t, T', m0) = mr(b-% 182
e{b, T'),
(4)
Short-Time Behaviors of Long-Ranged Interactions
2035
with X" = T/Tc. The characteristic function
(5)
The function e(6, T") is the solution of the following equation ek(b,T') l-e(b,T')
=6_2(fc_1)
T^_ 1-T'
(6)
Where k = d/a and z = a. At the critical point e(6,1) = 1. As mo —> 0, • bz/2mo, one attains m(i) ^ m o t 1 - *
(7)
That is ff = 1 - jfe/2. By use of the characteristic functions, the response propagator and correlation function have the generalized scaling forms which are valid for the times larger than t m ; c and for an arbitrary mo, Gp(t,f,mo)=p-2+"+^(pC(t),p$(t'),c(p-1,r'),¥'(p-1,mo))
Cp(tJ,m0)=p-2^g(pZ(t),pZ(t'),e(p-\T%
(8)
(9)
where the domain size12 £(i) ~ tp with p = 1/cr, h and g are two universal functions. 2. Dynamic Ginzburg—Landau m o d e l The Ginzburg-Landau model with anisotropic cubic term has a hamiltonian H
w-/^|f(v S ) 2 +^(v^) 2 +^ 2 +|( S 2 ) 2 +|f:(o 4 } (io)
where s — (sa) are n-component order parameter fields; gi and ga are the coupling constants for the isotropy and the anisotropy respectively. The SRI model corresponds to a = 1 and b = 0, whereas for the pure LRI model a < 2, a = 0 and 6 = 1 . The long-time relaxation behavior of the model has been extensively studied. 10 ' 13-15 Here we will concentrate on the short-time behavior. The dynamics is given by the Langevin equation
where A is the kinetic coefficient. The random forces £ = (£ a ) are assumed to be Gaussian distributed. The anisotropic cubic fixed point is stable when n > nc = 4 — 2Dae where Da = ip(l) — 2ij){a/2) + ip(cr) with ip(x) being the logarithmic derivative of the gamma function. In the two-loop level, we attained for n> nc 3(771
(_
n 2 - 1 8 n + 24 6n 2 183
2(71 + 2),, „ 3
(12)
2036
H. Fang, et al. P u r e P.P. n > 4, S < Si n+'l 2
SRQI F.P. n == 1, S < S3 1 < n < 4, 5 < <52 0 4CT
n ^ 1, (5 > <5i or 52 2(n-l)p+3ne 2
LRQI F.P. n = 1, 6 > 53
S^e1'2
n = 1,
e
0
6CT
Here we have introduced
B K
[i+x<J+{e+ n
^ ^ S^
^
with e a unit vector in the 2o--dimensional space, Kia = 21~2°/[n'7T(a)}. For n < nc one has the same result as in the isotropic model (ga = 0 and a < 2 — risr, with r]sr the Fisher exponent at the SRI fixed point) Vs+Va+ 2z e(n + 2) 2o{n + 8)
& = -
V0 1 +
7n + 20
[(n + 8)2
12(^2-0-5^)' a(n + 8)
(13)
A discussion on the weakly long-range regime of 2 — r)sr < a < 2 is given in Ref. 8. 3. D y n a m i c Ginzburg-Landau model with impurities The Hamiltonian with both LRI and long-range quenched impurities (LRQI) is defined as 2 2 H[S] = | ^ { i ( V t S ) 2 + ^ 2 + |4!' f(S ) +
V
(14)
The static random-impurity noises <j){x) describe the quenched disorders (random temperatures ) and satisfy the following configurational averages
{ 4 m(t) = mo (In — 184
(15)
Short- Time Behaviors
of Long-Ranged Interactions
2037
(2) the short-range disorder phase of 1 < n < 4 (4-n)
m(i)=m0(^J
n+2
(hi-J
(16)
(3) Ising short-range disorder of n = 1. In this case, the one-loop exponents are non-universal due to the degeneracy of /^-function.6 The two-loop result is m(t) ~ mo exp j ( ^ - )
^
In the above three equations, io>*d>*o
[ ( l n ^ ) ) 1 ' 2 - (Ht/t'>))1/2] an
|
(17)
d *i>*2 are microscopic time-scales
4. Monte Carlo simulations Monte Carlo simulation for 1-d LRI Ising model is being done. 17 The preliminary results are encouraging. For a = 0.7, the simulation gives 9' = 0.1648. This value agrees with our theoretical prediction 9' = 0.1673. Contrast to this, the theoretic values of 9' of the SRI are 0.131 for d = 3 and 0.356 for d = 2, whereas the results of Monte Carlo are 0.104 and 0.191 respectively. We also simulated a one-dimensional adsorption-desorption process (ADP) which is an irreversible non-equilibrium model. Each site could be either occupied(denoted by Si = 1) or not-occupied(denoted by Si — 0). The possibility of adsorption is A (provided the site under consideration is vacant). The desorption has a long-ranged correlated probability w(8i = l->si
= 0)=A
£ (1-sj) li-i|>o
) lJ
(18)
l|
By choosing A, the desorption probability is normalized to unity for the state that all sites are vacant. The critical exponent 9' versus the interaction range parameter a is plotted in Figure 1. Within the statistical error, one sees that the exponent of a > 2 has the same value as that of the short-range model. 18 In the last reference, the characteristic function is also discussed. References 1. H. K. Janssen, B. Schaub, and B. Schmittmann, Z. Phys. B73, 539 (1989) 2. B. Zheng, Int. J. Mod. Phys. B12, 1419(1998) 3. Y. Chen, S.H. Guo, Z.B. Li, S. Marculescu, and L. Schiilke, Eur. Phys. J. B18, 289(2000) 4. Y. Chen and Z.B. Li, J. Phys. A:Math. Gen. 34 1549 (2001) 5. Y. Chen, Phys. Rev. B 63, 092301 (2001) 6. Z.B. Li, Y. Chen, and S.H. Guo, Mod. Phys. Lett. B 15, 43 (2001) 7. Y. Chen, S.H. Guo, and Z.B. Li, Phys. Stat. Sol. B 223, 599 (2001) 8. Y. Chen, S.H. Guo, and Z.B. Li, Commun. Theor. Phys. (Beijing) 35, 475 (2001) 9. Y. Chen, S.H. Guo, Z.B. Li, and A.J. Ye, Eur. Phys. J. B 15, 97 (2000) 10. G. S. Joyce, Phys. Rev. 146, 349(1966) 185
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0.35-1
T
r
0.30-
I
*
11
0.2511
0.20-
0.15-
short-range
II
1.5
2.5
Fig. 1. Monte Carlo results of 8' of the one-dimensional absorption-desorption process for various values of a. The straight line is the result of the short-ranged interaction model. 11. 12. 13. 14. 15. 16. 17. 18.
B. Zheng, Phys. Rev. Lett. 77, 679 (1996) A. J. Bray, Adv. Phys. 43, 357 (1994) M. E. Fisher, S. Ma, B. G. Nickel, Phys. Rev. Lett. 29, 917 (1972) J. Sak, Phys. Rev. B 8, 281 (1973) J. Honkonen, J. Phys. A 23, 825 (1990) A. Weinrib and B. I. Halperin, Phys. Rev. B 28, 6545 (1983) Y. Chen, S.H. Guo, Z.B. Li, S. Marculescu, and L. Schueike, in preparing. Z.B. Li, S.P. Seto, M.Y. Wu, H. Fang, C.S. He, and Y. Chen, preprint LiOlb, 2001, Zhongshan University
186
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2039-2046 © World Scientific Publishing Company
GENERALIZED SUPERSYMMETRIES A N D C O M P O S I T E STRUCTURE IN M - T H E O R Y *
JERZY IAJKIERSKI Institute for Theoretical Physics, University of Wroclaw pi. M. Borna 9, 50-205 Wroclaw, Poland
Received 21 December 2001
We describe generalized D = 11 Poincare and conformal supersymmetries. The corresponding generalization of twistor and supertwistor framework is outlined with 0 5 p ( l | 6 4 ) superspinors describing BPS preons. The ^ BPS states as composed out of n = 32 — k preons are introduced, and basic ideas concerning BPS preon dynamics is presented. The lecture is based on results obtained by J.A. de Azcarraga, I. Bandos, J.M. Izquierdo and the author. 1
1. Introduction M-theory has been proposed as a hypothetical quantum theory describing elementary level of matter, which should incorporate and possibly explain various properties of "new string theory" (for review see e.g. Refs. 2,4). One of the features of such new theory of fundamental interactions should be the appearance of many extended elementary objects (p-(super)branes, D-(super)branes etc.) related with each other via duality/dimensional reductions net. Such a variety of basic objects in the theory makes sensible a search for some underlying composite structure. The basic dynamical degrees of freedom in M-theory yet are not known—there were presented only some proposals usually related with D = 11 space-time geometry. We postulate that the composite structure of M-theory should be formulated in terms of new degrees of freedom related with new geometry. Because M-theory is supersymmetric, and supersymmetry reveals more elementary nature of spinorial objects, we shall postulate that the basic fundamental geometric structure in M-theory is spinorial. The only well-known part of the description of M-theory is algebraic. Assuming that M-theory lives in D = 11 ( this assumption is consistent with description of D = 11 SUGRA as the low energy limit of M-theory) we can postulate the following
'Supported by KBN grant 5PO3B05620 187
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J.
Lukierski
basic D = 11 M-superalgebra a
{Qr, QS} = zrs(rvC)rspk + C T k l c ) r s 2 H + (r[Ml_Ma]c)rs Z^-M .
(i)
where /x, f — 0 , 1 , . . . 10, r,s = 1 , . . . 32. The collection of 528 Abelian generators Zrs {Zrs = ZST) describes the generalized momenta in M-theory. Introducing dual generalized coordinate space
xrs = (r„c) rs x» + ( r M c ) „ x^
+ ( i W M c ) r s x^-^,
(2)
we obtain large generalized phase space, with coordinates and positions described by the adjoint representations of Sp(32) algebra. Let us recall the assumption of Penrose twistor formalism in D = A8,11 that basic spinorial degrees of freedom in twistorial theory of elementary particles are described by N twistors (i = 1 . . . N) tW = ( A « , < ^ ) ,
(3)
where A ^ \ u^A (A = 1,2) are the pairs of D = 4 Weyl spinors. The following formula for the composite four-momentum is assumed 9 ' 11
^ = X>K\ where PAg = ^AB^ized momenta
(4)
We shall propose analogous formula in D = 11 for generalN
Z„ = £;A«AW,
(5)
i=l
where Ar (r = 1 . . . 32) are D = 11 real Majorana spinors. In D = 4 the twistors (3) are the fundamental representations of the spinorial covering SU(2, 2) of D = 4 conformal algebra (SU(2,2) = 5 0 ( 4 , 2 ) ) . In D = 11 there exists only minimal conformal spinorial algebra 12 ' 14 describing the classical real algebra Sp(64), containing D = 11 conformal algebra 50(11,2) C 5p(64; i?).
(6)
In Sect. 2 we shall consider the generalization of D = 11 Poincare and conformal superalgebras, supersymmetrizing the minimal D = 11 conformal spinorial algebra. In Sect. 3 we shall introduce in D = 11 the generalization of twistor and supertwistor formalism, with the extensions of Penrose-Ferber relations, which relate 05p(l|64) supertwistor space described by real coordinates (£2 = 0; R = 1 . . . 65)
TR = (\r,u>r,0, a
(7)
T h e relation (1) is the standard, minimal M-superalgebra. One can also add arbitrary spin-tensor central charges (see e.g. Refs. 5,6). The most general case was considered by Sezgin.7 188
Generalized Supersymmetries and Composite Structure in M-Theory
2041
with the generalized phase space (Xrs, Prs) (see (1-2)). In Sect. 4 we shall describe algebraically -^BPS states by n — 32 — k superspinors (7) representing D = 11 generalized supertwistors. These supertwistorial constituents we shall call BPS preons. It appears that our model geometrically corresponds to new type of KaluzaKlein theory, with discrete internal extension of space-time coordinates. 2. D = 11 Conformal M-(Super)Algebra Let us observe that the D = 4 conformal algebra (P^, M^„, D, K^) is endowed with the following three-grading structure L\
LQ
L_I
(8) Grading (8) in determined by the scale dimensions of generators [ D , i y = *M>
I A M J = 0,
[D,Ktl] = -K»
(9)
and it is easy to see that the conformal algebra (8) has two Poincare subalgebras: (Pn,M^) and (K^M^). For D = 4 superconformal algebra 577(2,2; 1) = (P^ M^v, D, A, K^; QA, QA, $A, SA) the three-grading (8) is extended to the following five-grading L\
L\j2
£-1/2 L-i
LQ
(10) P„
QA, QA M^V,
D, A
SA, SA
K„
where consistently [D,QA]
= \QA,
[D,SA\
[D,QA\ = \QA,
= -\S
A
,
[D,SA] = \SA,
and again SU(2,2; 1) contains as subsuperalgebras the Poincare superalgebras (P^, M^v,
QA; QA)
a n d
(K^M^SA^J).
The structure of D = 11 generalized superconformal algebra, which we call conformal M-superalgebra is quite analogous. The D = 11 conformal M-algebra Sp(QA) can be in analogy to (8) described by the following three-grading
Z/rs
**-rs
^rs
528 Abeiian GL(32; R) 528 generators
algebra
(12)
Abelian
generators
We see that Sp(64) contains two copies of generalized D = 11 Poincare algebras, described by inhomogeneous Sp(32) algebras (Sp(32; R) C GL(32;R)) with 528 Abelian translation generators. 189
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J. Lukierski
The superextension of D = 11 conformal M-algebra 0 5 p ( l ; 6 4 ) which we call conformal M-superalgebra is described by the following five-grading (see also Refs. 15,16) L\
L1/2
LQ
£*TS
St 7"
" T S
£-1/2
L-i
^7*
"7"S
(13) where (Qr, Sr) are the pair of 32-component supercharges, transforming as fundamental representations of Sp(32), with RrS C Sp(32) if RrS = Rsr. The subalgebras spanned by the generators (Qr, Zrs) and (Sr, Zrs) describe two copies of M-superalgebra given by the relations (1). It should be added that the gradings (12,13) correspond to the grading structure of real Jordanian (super) algebras. 17 ' 18 3. D = 11 Supertwistors and Their Relation with Generalized Superspace Let us recall two basic relations of Penrose twistor theory in D = 4 8 ' n (i) relation between the generators of Poincare algebra and twistor components (2) PAB MAB
=\
A WB)
(14)
= *A\B,
,
MAB
= ~\(A u)B)
(15)
where MAB = \{a^v)ABM'lu and MAB = ^ ( v J ^ M ' " ' 1 ' . The relations (15) can be extended to all 15 generators of D = 4 conformal algebra, (ii) Penrose incidence relation between twistor and space-time coordinates u>A = i\BXBA
UJA = -iXABXB
(16)
where XBA = (XAB)* describe four real Minkowski coordinates if the SU(2,2) twistor norm vanishes (t,t)=i(\Au>A-\Au>A)=Q.
(17)
The relations (14-17) can be supersymmetrized. If we introduce the D = A supertwistor (ta,r)), which is the fundamental representation of SU{2,2; 1) with complex Grassmann variable rj (r/2 = rf = 0, {r/, rj} = 0), the relations (14-15) has been extended by Ferber 19 to all generators of D = 4 superconformal group SU(2,2;1). The Penrose relations (16-17), firstly supersymmetrized in Ref. 19 look as follows b
We recall that {O^)AB = M^^AB^^B
~ (<7")AB^B1
190
=
- § < ^ " P T ( < T P T ) A B = [(07,1/)^]*
Generalized Supersymmetries
and Composite Structure in M-Theory
2043
wA = i\B ZBA = i\B (XBA - ieBeA) uJA=i(xAB+i6A0B)\B V = XAOA
rj=XA6
.
(18)
For D = 11 the generalized twistors and supertwistors are real (see (7)) and the real OSp(l; 64) superalgebra (R, S = 1... 64) {QR,QS}
= RRS,
(19)
QR = -i= TRZ,
(20)
can be obtained if we assume that c . RRS
=TRTS
where TR describes D = 11 real twistorial quantum phase space the 5^(64) antisymmetric metric) [TR,TS]
(TJRS
=
—TJSR
= iriRS ,
is
(21)
supplemented with trivial one-dimensional Clifford algebra relation £2 = 1. The relations (19) are extended to D = 11 as follows: wr = {Xrs - iOT 8s) \ s
£ = 9r\r.
(22)
Relations (22) relate the D = 11 supertwistor space coordinates (7) with the extended D = 11 superspace (Xrs,6s), described by 528 bosonic and 32 fermionic coordinates. 4. B P S States in M - T h e o r y and Composites of BPS Preons The -^BPS state |fc) can be denned as an eigenstate of generalized momenta generators Zrs\k) — zrs\k),
(23)
such that det zrs = 0. The number k determines the rank of generalized momenta matrix zrs k — B P S state: {rank zrs = n = 32 - k;
1 < k < 32} .
(24)
^.Frorn (24) follows that the BPS state \k) preserves a fraction v = ^ of supersymmetries. c
By Bott periodicity this realization is related with twistor framework in D = 3 (see Ref. 20), also with real structure. In D = 5, 6, 7 one has t o use the extension of Penrose framework t o quaternionic twistors (see e.g. Ref. 21 for D = 6). 191
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We call BPS preon the hypothetical primary object carrying the following generalized momenta 1 Zirs = Xr A S .
\4o)
The formula (25) corresponds to putting n = 1 in the relation (5) and describes ^BPS state. More general formula (5) describes the generalized momenta of a system composed out of n BPS preons and it describes (for 1 < n < 32) the ^BPS state (we recall that k - 32 - n). The number n = 32 — k of zero eigenvalues of the matrix zrs determines the number of independent supercharges Qr , annihilating the BPS state |fe). These supersymmetries, preserving the BPS state, are called in p-brane theory the Ktransformations. We see that the supersymmetric D = 11 single BPS preon dynamics should have 31 K-symmetries. Recently 22 such dynamical superparticle models d with fundamental OSp(l;2n) superspinor as basic variable has been proposed. It should be recalled here (see e.g. Ref. 24) that in the standard super p-brane formulations half of the supersymmetries are promoted to K-transformations, i.e. in D = 11 we obtain 16 /^-transformations. Using the D = 11 supertwistor description with the relations (22) and (25) providing a bridge between BPS preons and generalized space-time, we can formulate three different geometric pictures: (i) Purely supertwistorial picture, with basic phase space parametrized by BPS preon coordinates T^' (see (7)). The canonical Liouville one-form describing free action is given by the relation n
fil = Y. {^^
dXi
r + £W dZ(i)) .
(26)
which can be supplemented by some algebraic constraints, (ii) Mixed geometric picture, with the components w ^ expressed by means of the relation (22). One obtains from (26) n
n2 = ] T A « r A<*> (dXrs - i6T d6s) ,
(27)
i=l
(iii) Generalized space-time picture, with the relation (5) inserted in (27). 0 3 = Zrs (dXrs - i9r d6s) .
(28)
The application of these three geometric pictures to the description of D = 11 dynamics (for n > 1) is under consideration. d
F o r D = 4,6 and 10 see Refs. 22,23. 192
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5. F i n a l R e m a r k s We mention here two interesting aspects of t h e presented approach which deserve further attention; (i) geometric confinement of BPS preons Because the space-time coordinates a r e composed out of preonic degrees of freedom, the D = 11 space-time point c a n b e determined only in t e r m s of at least 16 preonic set of spinorial coordinates. This is t h e D = 11 extension of known property of Penrose theory in four dimensions w i t h two twistors needed for t h e definition of composite Minkowski space-time points. (ii) internal symmetries T h e formula (5) expresses 528 generalized m o m e n t a in t e r m s of 32n preonic spinorial coordinates A , (i = 1, . . . n ) . T h e internal symmetries can b e obtained by interchanging BPS preons. For t h e case n = 1 6 corresponding to the choice of v = ^SUSY one can introduce internal 0 ( 1 6 ) symmetries, leaving t h e values of Zrs invariant.
Acknowledgements T h e a u t h o r would like t h a n k prof. Mo-Lin G e for his w a r m hospitality a t Nankai Symposium in Tianjin.
References 1. LA. Bandos, J.A. de Azcarraga, J.M. Izquierdo and J. Lukierski Phys. Rev. Lett. 86, 4451 (2001). 2. M.J. Duff, R.R. Khuri and J.X. Lu, Phys. Rep. 259, 213 (1995). 3. P.K. Townsend, hep-th/97120044. K.S. Stelle, hep-th/9803116 . 5. E. Bergshoeff and E. Sezgin, Phys. Lett. B 2 3 2 , 96 (1989); B 3 5 4 , 256 (1995). 6. C. Chrissomalakis, J.A. de Azcarraga, J.M. Izquierdo and J.C. Perez Bueno, Nucl. Phys. B567, 293 (2000). 7. E. Sezgin, Phys. Lett. B392, 321 (1997). 8. R. Penrose and M.A.H. MacCallum, Phys. Rep. 6, 241 (1972). 9. Z. Perjes, Phys. Rev. D l l , 2031 (1975). 10. R. Penrose, Rep. Math. Phys. 12, 65 (1977). 11. L.P. Hugston , "Twistors and Particles" , Lecture Notes in Physics, Vol 97 (Springer Verlag, Berlin, 1979). 12. S. Ferrara, Fortschr. Phys. 49, 485 (2001). 13. R. D'Auria, S. Ferrara, M.A. Lledo, V.S. Varadarajan J. Geom. Phys. 40, 101 (2001). 14. R. D'Auria, S. Ferrara, M.A. Lledo, Lett. Math. Phys. 57, 123 (2001). 15. I. Bars, Phys. Lett. B457, 275 (1999); B 4 8 3 , 248 (2000). 16. O. Baerwald and P. West, Phys. Lett. B 4 7 6 , 157 (2000). 17. M. Cederwall, Phys. Lett. B115, 389 (1988). 18. M. Gunaydin, Mod. Phys. Lett. A 1 5 , 1407 (1993). 19. A. Ferber, Nucl. Phys. B132, 55 (1978). 20. W. Heidenreich and J. Lukierski, Mod. Phys. Lett. A5, 439 (1990). 193
2046
J. Lukierski
21. J. Lukierski and A. Nowicki, Mod. Phys. Lett. A6, 189 (1991). 22. I. Bandos, J. Lukierski and D. Sorokin, Phys. Rev. D 6 1 , 045002 (2000). 23. I. Bandos, J. Lukierski, Ch. Preitschopf and D. Sorokin, Phys. Rev. D 6 1 , 065009 (2000). 24. D. Sorokin, Phys.Rep. 329, 1 (2000).
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International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2047-2053 © World Scientific Publishing Company
GAUSSIAN O R T H O G O N A L E N S E M B L E FOR THE LEVEL SPACING STATISTICS OF T H E Q U A N T U M FOUR-STATE CHIRAL POTTS MODEL
J.-Ch. A N G L E S d'AURIAC and S. DOMMANGE 25 Avenue
CRTBT, CNRS, BP 166X des Martyrs, 38042 Grenoble Cedex, France
J.-M. MAILLARD and C. M. VIALLET LPTHE, Tour 16, ler etage, boite 126, 4 Place Jussieu, 75252 Paris Cedex 05, France
Received 25 October 2001
We have performed a R a n d o m Matrix Theory (RMT) analysis of the quantum four state chiral Potts chain for different sizes of the quantum chain up to eight sites, and for different unfolding methods. Our analysis shows that one generically has a Gaussian Orthogonal Ensemble statistics for the unfolded spectrum instead of the GUE statistics one could expect. Furthermore a change from the generic G O E distribution to a Poisson distribution occurs when the hamiltonian becomes integrable. Therefore, the RMT analysis can be seen as a detector of "higher genus integrability".
Introduction : the quantum chiral Potts chain Since the pioneering work of Wigner 1 and Dyson, 2 Random Matrix Theory (RMT) has been applied successfully in various domains of physics. One motivation is to describe, in a united universal framework, various phenomena implying chaos3 or at least complexity. An extreme case is the emergence of integrability which manifests itself in the drastic change of the generic wignerian energy level spacing distribution into poissonian distribution. The first examples of this connection emerged when one considered simple harmonic oscillators or free fermions models. This reduction to Poisson distribution reflects nothing but the independence of the eigenvalues. At this point it is natural to ask whether this link between Poisson reduction and YangBaxter integrability still holds when the solutions of the Yang-Baxter equations are no longer parametrized in terms of abelian varieties. The perfect example to address this question is the chiral Potts model for which Au-Yang et-al have found a higher genus Yang-Baxter solution. 4 The Hamiltonian of the quantum chiral Potts chain first introduced by Howes, Kadanoff and den Nijs 5 and also by von Gehlen and 195
2048
J.-Ch. Angles d'Auriac et al.
Rittenberg 6 is defined as : JV-1
H^Hx + Hzz = YlHjj+i = -J2Il[^-(XJr j
j
+ an-(ZjZ}+ir}
(1)
n=l
where Xj = IN ® • • • ® X <8> • • • ® IN and Zj = IN®---®Z<8>---® IN- Here IN is N x N unit matrix, while X and Z are N x N matrices whose elements are defined by Z^m = 5j,m exp[27ri(j — l)/iV] and Xjitn = Jj, m +i (mod N). The self-dual model 7 corresponds to an = a^. Some spectral analysis of this model have been performed for the quantum self-dual model or the 3-state model. 6 ' 8 In this paper we examine the N = 4 (four state chiral Potts model) non self-dual case. The conditions for the quantum Hamiltonian to commute with the transfer matrix family (integrability conditions) read (see equations (33a), (33b), (33c) and (33d) in Ref. 9) : al2+al2 = a\ + <*§ oj a.2 2 2 {of - o^ )(2o^ - a^) =0
of = al aict3 aids ' {a\ - al)(2a\ - « i a 3 ) = 0,
(2)
In order to have a real spectrum we also choose to have an hermitian hamiltonian restricting to conditions ai = 03 and a~\ = 03* (where the star denotes the complex conjugate). A possible parametrization is then : ot\ = a% = y/l + r + iy/1 - r, 2
2
ai" = 03* = y n + rn + i\/n
a2 = 1 —rn,
(3)
a^ — n
where r and n are real. The value n = 1 corresponds to the self dual situation. 1. The R M T machinery. RMT analysis considers the spectrum of the (quantum) Hamiltonian, or of the transfer matrix, as a collection of numbers, and looks for some possibly universal statistical properties of this collection of numbers. Obviously, neither the raw spectrum, nor the raw level spacing distribution, have any universal properties. In order to uncover universal properties, one has to perform some normalization of the spectrum: the so-called unfolding operation. This amounts to making the local density of eigenvalues of the spectrum equal to unity everywhere3. In other words, one subtracts the regular part from the integrated density of states: one considers only the fluctuations. It has been found that the unfolded spectra of many quantum systems are very close to one of four archetypal situations described by four statistical ensembles. For integrable models this is the statistical ensemble of diagonal random 3
T h e unfolding can be achieved by different means. Let us note however that there is no rigorous prescription and the "best criterion" is the insensitivity of the final result to the method employed or to t h e parameters (for "reasonable" variation). A detailed explanation and tests of these methods of unfolding are given in Ref. 10. 196
Gaussian Orthogonal Ensemble for the Level Spacing Statistics
2049
matrices and the level spacing distribution is close to a Poissonian (exponential) distribution, P(s) = exp(—s). For non-integrable systems it can be the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE), or the Gaussian Symplectic Ensemble (GSE), depending on the symmetries of the model under consideration. If the hamiltonian is time reversal invariant 13 the level spacing distribution is either described by the Gaussian Orthogonal Ensemble (GOE), or by the Gaussian Symplectic Ensemble (GSE): P
G0E(*)
= |sexp(-7rS2/4),
PGSE(S)
= B3s4exV(-Bs2)
(4)
where B = ( | ) ^ ~ 2.263. Note that GOE can also occurs in a slightly more general framework ("false" time-reversal violation, A-adapted basis 12 ). When one does not have any time-reversal symmetry (or "false time-reversal symmetry") the Gaussian Unitary Ensemble distribution should appear :
PGm(s) = ^s2eM-^2M
(5)
To quantify the "degree" of level repulsion, it may be convenient to use a parametrized distribution which interpolates between the Poisson law and the GOE Wigner law. Among the many possible distributions we have chosen the Brody distribution: Pf}(s)= (l + /3)c 2 s / 3 exp(-c 2 s / 3 + 1 ),
1.1. Representation
with
c2 =
/3 + 2 /3 + 1
-«-TM
(6)
theory
In the presence of symmetries, one should distinguish eigenstates according to their quantum numbers. This is an essential requirement of the method. For instance both lattice shift and shift of colour commute with the hamiltonian H. They generate a symmetry group S = ZL (g> Z4 which does not depend on the parameters cti,ai of the hamiltonian H. Since the group 5 = ZL ® Z4 is abelian one may diagonalize simultaneously all the elements of the group S as well as the hamiltonian H on the S-invariant spaces. This amounts to block-diagonalizing H and to split the spectrum of H into the many spectra of each block. The construction of the projectors is done with the help of the character table of irreducible representations of the symmetry group. Details can be found in Ref. 10 and Ref. 14. In this work we concentrate on the four-state case (N = 4) of the quantum hamiltonian (1). For generic r and n in parametrization Eq. (3), the total symmetry group is ZL ® Z4. Since the characters of Z^ (g> Z\ are complex, one has to use complex numbers even though the final results are real, which increases the programming difficulties. We always restricted ourselves to hermitian hamiltonians. Consequently the blocks are also hermitian and there are only real eigenvalues. The diagonalization is performed using standard methods of linear algebra (contained in the LAPACK library). 197
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J.-Ch.
Angles d'Auriac
et al.
2. Results. We show in this paper that the RMT analysis can act as an integrability detector. More specifically we want to exhibit the transition to integrability when the parameters meet conditions Eq.(2). We thus choose to move in the a*, 57 parameter space along a trajectory compatible with the hermiticity of the hamiltonian, generalizing the parametrization (4) and crossing the integrable variety Eq.(2) : a.\ = a% = y/l + r + i\J\ —r , 2
2
o l = 03* = \ / n + rn + iyn
a2 = t — rn,
02= n
(7)
where t, r and n are real, with n / 1 to avoid the self-dual case. The value a2 = t = 1 thus corresponds to the occurrence of genuinely "higher genus integrability" on this trajectory. We have constructed the quantum Hamiltonian (1), of the four state Potts model (1), for various linear sizes, up to size eight (L = 8), leading to matrices of size up to 4 8 x 4 8 = 65536 x 65536. The results, displayed below, show that the size L = 8 is sufficient to answer the question we addressed. Using the complex characters and projectors associated with the group Zi ® Z 4 we have performed the block diagonalization of the hamiltonian. The sum of the dimensions of all the blocks corresponding to the 8 x 4 = 32 representations, is 4 8 = 65536 as it should. We then performed the unfolding in each block independently. The behavior in the
L=8, R-(0,0) n-2.1, r-0.5, t-1.5, b=0.99 T
Fig. 1.
1
1
1
r
Level spacing distribution versus GOE, GUE, GSE and Poisson. 198
Gaussian Orthogonal Ensemble for the Level Spacing Statistics
2051
various blocks (representations) is not significantly different. We also compared four different unfolding procedures, again getting similar results. We display the results on the largest size L = 8 for the best unfolding procedure, namely the gaussian unfolding. Figure 1 shows the level spacing distribution P(s), for the representation (0,0) and for r = 0.5, n = 2.1, and t = 1.5, which corresponds to ct\ = a^ = 1.225 + i 0.707, a2 = t = 1.5, en = 03* = 2.337 + i 1.833 and o£ = 2.1. This figure shows the energy level spacing distribution and the corresponding brody fit (6) for the (least square) best value found to be fibrody = 0.99. On the same figure the GOE level spacing distribution is also displayed, both curves are almost indistinguishable. The GUE or GSE level spacing distribution are clearly ruled out, as well, of course, as the Poisson distribution. Very similar results are obtained for all the distributions corresponding to the other representations and other values away from the integrability value #2 = t = 1. Let us now consider the (higher genus) integrable case which corresponds, with our parametrization, to a
L=8, R=(0,0) n=2.1, r=0.5, t=1,(J=0.04
Fig. 2.
Level spacing distribution on the integrability variety.
Figure 2 displays the level spacing distribution, compared to a Poisson distribution (and also to the GOE level spacing distribution), for the integrable case 199
2052
J.-Ch. Angles d'Auriac et al.
r = 0.5, n = 2.1, and t = 1 which corresponds to cti = a\ = 1.225 + i 0.707, a2=t = l, Q7 = 5 J * = 2.337 + i 1.833 and a j = 2.1. The best brody distribution approximation of the data is found to be for Pbrody = 0.04 using a least square fit. We have obtained very similar results (namely an extremely good agreement with a Poisson distribution) with other values of the parameters n and r, and for the various representations, when t is kept equal to the (higher genus) integrability value t = 1. The RMT analysis can therefore be used to detect integrability even when the integrability is not associated with abelian curves but is a more subtle integrability where higher genus curves occur. This extremely good agreement with an independent eigenvalues framework is found for t = 1 exactly. When t is slightly different from 1, one is clearly no longer Poissonian in agreement with the fact that the Poissonian framework should only correspond to the integrable value t = 1. In order to quantify the (finite size) transition from integrability to chaos, we calculate the best brody parameter, as a function of the parameter t, keeping r and n constant. Figure 3 displays (3brody, as a function of t, for all the representations.
oi
i
' 0.2
i
i
i
i
i
i
i
0.4
0.6
0.8
1
1.2
1.4
1.6
Fig. 3.
1.8
The brody parameter, as a function of the parameter t.
These results confirm a quite sharp transition from a GOE distribution to a Poisson distribution. In the thermodynamic limit one can expect fibrody to be equal 200
Gaussian Orthogonal Ensemble for the Level Spacing Statistics
2053
to the G O E value fibrody — 1 for every value of t h e parameter t, except at point t = l, where t h e Poisson value Pbrody = 0 should occur.
References 1. E. P. Wigner, Ann. Math. 62, 548 (1955); 65, 203 (1957); 67, 325 (1958) 2. F. J. Dyson, J. Math. Phys. 3, 140 (1962). F. J. Dyson, J. Math. Phys. 3, 157 (1962), F. J. Dyson, J. Math. Phys. 3, 166 (1962). 3. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Interdisciplinary Applied Mathematics, Springer-Verlag, New York Berlin Heidelberg, 1990. 4. R. J. Baxter, J. H. H. Perk and H. Au-Yang, Phys. Lett. A 128, 138 (1988) 5. S. Howes, L. P. Kadanoff and M. den Nijs, Nucl. Phys. B 215, 169 (1983). 6. G. von Gehlen and V. Rittenberg, Nucl. Phys. B 257, 351 (1985). 7. L. Dolan and M. Grady, Phys. Rev. D 25, 1587 (1982). 8. G. Albertini, S. Dasmahapatra and B. M. McCoy, Exact Spectrum of the S-state Potts Chain, in Yang-Baxter Equations in Paris, edited by J-M. Maillard (World Scientific, 1992) 9. H. Au-Yang, B. M. McCoy, J. H. H. Perk, S. Tang and M-L. Yan, Phys. Lett. A 123, 219 (1987) 10. H. Bruus and J.-C. Angles d'Auriac, cond-mat/9610142 (1996). H. Bruus and J.-C. Angles d'Auriac, Phys. Rev. B 55, 9142 (1997). 11. D. Poilblanc, T. Ziman, J. Bellisard, F. Mila, and G. Montambaux, Europhys. Lett. 22, 537 (1993). 12. M. Robnik and M. V. Berry, J. Phys. A 19, 669 (1986). 13. E. P. Wigner, Group Theory and its application to the quantum mechanics of atomic spectra, (Academic Press, New-York and London, 1959). 14. H. Meyer, Ph.D. thesis, Univ. J. Fourier, Grenoble, France, 1996.
201
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2055-2064 © World Scientific Publishing Company
COMMENTS ON T H E D E F O R M E D WN
ALGEBRA*
SATORU O D A K E t Department of Physics, Faculty of Science, Shinshu Matsumoto 390-8621, Japan
University
Received 15 November 2001 We obtain an explicit expression for the defining relation of the deformed Wjy algebra, DWA(s[jv)g,t- Using this expression we can show t h a t , in the q —> 1 limit, DWA(sljv)<3,t with t = e
_ 27TX fe + JV
*r q~H"~ reduces to the sl^r-version of t h e Lepowsky-Wilson's ^-algebra of
-~
~.
_ 2-rri
k+N
level k, ZA(slN)k- m other words DWA(sljv)g,t with t = e N q N can be considered as a g-deformation of ZA(sljv)fc. In the appendix given by H. Awata, S. Odake and J. Shiraishi, we present an interesting relation between DWA(s[jv)g,t and ^-function regularization.
1. Introduction One of our motivation for study of elliptic algebras (deformed Virasoro and W algebras, elliptic quantum groups, etc.) is to clarify the symmetry of massive integrable models. Massive integrable models includes quantum field theories with mass scale and solvable statistical lattice models. Typical examples of the latter are models based on s ^ : Andrews-Baxter-Forester (ABF) model and Baxter's eight vertex model. About these models we know the following:1 model
ABF (III)
8 vertex
Boltzmann weight
face type
vertex type
algebra gradation (energy level of He) space of states free field realization
B«,A(al2)
( B ® { P , e « } = C7,,p(«[2))
Aq,p(sl2)
homogeneous gradation
principal gradation
irr. rep. space of DVA direct (construction of VO)
irr. rep. space of Aq,p(sl2) indirect (map to ABF)
"Talk at the APCTP-Nankai Joint Symposium on "Lattice Statistics and Mathematical Physics", 8-10 October 2001, Tianjin China. [email protected] 203
2056
5 . Odake
In order to obtain more direct free field realization of the eight vertex model and its higher rank generalization, it may be useful to study (deformed) current algebras of sljv in principal gradation. Motivated by this, Hara et al.2 studied free field realization of the Lepowsky-Wilson's Z-algebra 3 and found some relation between the deformed Virasoro algebra (DVA) and .Z-algebra. Recently Shiraishi constructed a direct free field realization of the eight vertex model with a specific parameter p = q3, where the type II vertex operator is given by the DVA current. 4 In this article we extend the relation between DVA and .Z-algebra to the higher rank case. In section 2 we present an explicit expression for the defining relation of the deformed WM algebra. This is a main result of this article. In section 3, by using this explicit expression, we show that the deformed W}v algebra reduces to the sljy-version of the Lepowsky-Wilson's Z-algebra in some limit. In the appendix given by Awata, Odake and Shiraishi, we present an interesting relation between the deformed WN algebra and ^-function regularization.5 2. Deformed Wjv Algebra 2.1.
Definition
Let us recall the definition of the deformed WN algebra, DWA(s[jv)g,t-6'7 It is defined through a free field realization. This algebra has two parameters(g and t), and we set t = q13 and p = qt_1. Let us introduce fundamental bosons hln (n 6 Z ; i = 1, • • •, N ; J2i=iPinK. = 0) which satisfy
K,HJ = - - ( i - gn)(i - t~n)
v_ pNn
Nne{i<j) P
sn+m,0,
(i)
where 0(P) = 1 or 0 if the proposition P is true or false, respectively. Exponentiated boson Ai(z) (i = 1, • • •, N) is defined by = : e x p ( £ htnz-n)
\t(z)
: q^h°p^-\
(2)
ra^O
Here : * : stands for the usual normal ordering for bosons, i.e., hln with n > 0 are in the right. By using this Aj(z), DWA(sljv)q,t current Wl(z) = E « e z ^ n z _ n (i = 1, • • •, TV — 1) is given by W*(z)=
£
:Ah(p^z)AJ2(p^Z)...Aji(p-^z):,
(3)
l<jl<J2<"-<ji<W
and we set W°(z) = WN(z) = 1. (Remark that Aj(z) corresponds to the weight of vector representation of sljv and Wl(z) corresponds to the i-th rank antisymmetric tensor representation.) DWA(sljv)g,t is defined as an associative algebra over C generated by W^. a a
I t is also defined as a commutant of the screening currents. 6. 204
Comments
on the Deformed WJV Algebra
2057
The highest weight state |A) is characterized by Wn\X) = 0 (n > 0) and W&\\) = to* (A) | A) (wl(X) G C), and the highest weight representation space is obtained by successive action of WLn (n > 0). Since DWA(slN)q,t has two parameters^ and t), we can take its various limit by relating q and t. In the following limit b T-
-*T
L a m t I :
J « = e,i > i t = gt>,
/i->0 /3:fixed
(ao = V 3 - ^ ) ,
(4)
DWA(s(jv)q,t reduces to the WN algebra with the Virasoro central charge c = (N — 1)(1 - N(N + l)ao) because the g-Miura transformation of DWA(sljv) becomes the Miura transformation of WN algebra. Each DWA current Wl(z), however, reduces to some linear combination of WN currents. 2.2.
Relation
In order to write down relations between DWA currents, we define the delta function 6(z) = 2 „ G Z zn and the structure function fl,:>{z) = Yle>o fl'*2* (1 ^ M ^ - N - l ) ,
It has been expected that DWA currents satisfy quadratic relations, P^{^-)W%{z\) W:>(z2) — W^{z2)W%{z\)PA{^-) = (terms containing delta function), in mode expansion it becomes
m, wi] = - £ fiJ (wLtwi+i - wi_twi+l) £>1
-(-(contribution from the terms containing delta function).
(6)
For i = 1 and j > 1 case, the relation is 6 ' 7
fl>i{%)W\zx)Wt{z2)
- W\Z2)W\Zl)P'\%)
(j > 1)
= -{1~f^~t~1\KP^%)W3+1(Ph2)-5(p-i¥%)WJ+\p-h2)),
(7)
and for i = 2 and j > 2 case, the relation is 7 / 2 J ( a ) W 2 ( Z l ) W ( z 2 ) - Wi(z2)W2(Zl)P'2(%) i-p
(
i_
g )
(i_ri)
I-P
U > 2)
(i-p)(i-P2)
•(5(pi%)0oW1(p-h1)Wj+1(piz2)l
b Usually we call this limit as a conformal limit. However there are many other limits in which the resultant algebras are related to conformal field theory.
205
2058
S. Odake
-s(p-^yow1(piz1)w^1(P-iz2yo) +(1
" t-pr~1)2 ( ^ ( r ^ ' + ' f r * ) + T^wi+2^)
^
-*(p"*S)(r^^+2(^) + i r ^ ^ V 1 * ) ) ^ Here a normal ordering for currents ° * ° is denned by SW*(rz)W"'(z)°
OO
= E
E
771
# ' ' (rm-eWimWl+m
E
+r'-»-1e
m
.1r
m + 1
) • z-»,
(9)
n€Z m = 0 ^=0
where y^r stands for ^ « > o z™- ^ u e *° this n o r m a l ordering, infinite sums in the RHS of (6) become finite sums on the highest weight representation space. Eqs.(7) and (8) are directly calculated by using the commutation relation of h%n. In principle, we can continue this calculation for i > 3 cases, but in practice it is hopeless. So we use another method: fusion and induction. To write down general formula, we extend the range(0 < i < N) of Wl{z) and that(l < i, j < N - 1) of P'j{z) toieZ and i , j e Z respectively ; W\z) = 0 for i < 0 or i > N, and f'j(z) is given by (5) for all i, j G Z. Explicit expression of the defining relation of DWA(s[jv)gj4 is as follows: f'j(^)Wi(z1)W\z2)
y
- W^(z2)Wi(z1)f'i(^)
fc=l
i+k
(0
J=l
k j+k
x(d(p^ fi)f- ' {p-i^)Wi-k(p-h1)Wi+k(ph2) -8{p-^+k^)f-k'j+k(p^i)wi-k(ph1)wj+k(p-h2)),
(io)
. i . (1 — qz)(l — i - 1 z ) . . . . where 7(0 2 z) = —777r—• We can rewrite the RHS of this relation in
(l-z)(l-pz)
terms of the normal ordering ° * ° by repeated use of the following formula, which is obtained from (9) and (10), fi'j(r-1)Wi(rz)Wj(z)
(0
(1
rl)
= iw*{rz)W{z)i+ 'f: En^) V
y
fe=i 1=1 i k i+h ^+k)f - ' (P-Lii)wi-k(p1^tJ1z)w^k(ph)
( _ ,1—rp l_^ 1 — rp
f i
-
W
^
) w
i-k
206
{ p
-^
z ) W J +
k
{ p
-!t
z )
\
{ n )
Comments on the Deformed WJV Algebra 2059 where r e C is a "good" number (such that it does not give poles, see (15)). For example, (8) is easily recovered by (10) with i = 2 and (11) with i = 1. In order to prove (10) we present some formulas. Direct calculation shows f1'i(p±'^z)fi'j(z) = f+^ip^z) f1'i(p±(^i+k)z)f1^(z) 1i
± i
= +i
f ' (p ^ z)f^(z)
x \ \ i=i±1 *<\ L7U> 2 z) t>j f1'i-k(P±i^z)f1'j+k(p±%z)
±
±
= f^ (p ^h(p h)
(j> 1), (12)
(i,j,i-k,j
+ k>l), (13)
( M > 1 ) ,
(14)
and fa'b in the RHS of (10) is regular. By computing {\\f'j {^)Wi{z1)W^ in the free field realization, we can show that (10) implies Poles of fi'j(^)Wi(zi)W^(z2) ZL=p±{^+k)
(0
{z2)\\)
are
(l
(15)
because (Xlf1^ {^)Wt(zi)W:> (z2)\X) is a Taylor series in ^-, and for any states of the highest weight representation space, \ip) and \<j>), (ip\f'l,:'(^)W"l(zi)WJ(z2)\4>) differs from {\\fh:'(fL)Wl(zi)W:>(z2)\\) only for finite number of terms (Laurent polynomials in z\ and z2), which do not create other poles. (See also Appendix C of Ref. 9 where different notation is used.) Therefore fa>bWaWb in the RHS of (10) is regular and we can reverse its order, fa>b(pc)WaWb = fb
{l-p±^^)f1'j(^)W1(z1)W^z2)
( l - g ) ( l - * ~ a ) ^ + l(p±lZ2)
^<j
(16)
and if (10) is correct, (10) implies ^ = ±(-l~ ^
(1
~P:fipf2)fJ'WWj(z2)Wi(z1)
~ * - 1 ) f[i(pl+li)
• Wj+i(p^Zl)
(0
(17)
Proof of (10) : (i) The case i = 0 and i < Vj < N, and the case j - N and 0 < Vi < j are trivial, (ii) The case i = 1 and i < Vj < N, i.e. (7), is already proved, (iii) Let us assume (10) holds for i(< N) and i < Vj < N. We will show (10) holds for i + 1 and i + 1 < Mj < N. (For i = N — 1, we have i + l = N<j = N. Therefore it is sufficient to consider i < N — 1 and j < N.) Multiply f1'i(^)fhj(^)W1(z3) from left to (10) with i > 1 (whose second ab c a b 3 f ' (p )W W fb'a(p~c)WbWa), t e r m i n the RHS is replaced by reversed order one rewrite / l j ' ( f f ) W 1 ( ^ ) W j ( z 2 ) = P ' H f )W^{z2)W1{z3) + - • • by using (7), multiply n - o U i - t - 1 ) ^ "~P - l T ~;r)> an( ^ t a ^ e a ^ r a ^ Z3 ~~^ P _ i ^ z i - By using (12)-(16) and 207
2060
S. Odake
(17) (with j —> j + 1), studying poles carefully and replacing z\ —tp^zi, we obtain (10) with i -* i + 1 (2 < i + 1 < j < N). (iv) Therefore we have proved (10) by induction on i. rj
3. Relation t o
Z-Algebra
Affine Lie algebra slff is an associative algebra over C with the Chevalley generators, ef and hi (i = 0 , 1 , • • •, N — 1), which satisfy [hi, hj] = 0,
[hu ef] = ±Oijef,
[ef,ej] = Sijhi,
(18)
and the Serre relation {&Aef)1~aiief = 0 (i ^ j), where (a,ij)o
,.._..
In current basis SIN is given as follows: homogeneous gradation generators : Hln, E„'1 {n £ Z, 1 < i < N — 1), k : center, d : grading operator. relations : [Hi, Wm] = kaijn6n+mt0,
[Hi E±>>] =
l
[E+-*, E~^] = S*(H n+m + knSn+m,0),
±aijE^m, [d, Xn] = nXn
(X = H\
E^),(20)
and [E„'%,E^] = [^ n l*i,-E m +i] and the Serre relations which we omit to write explicitly, where (a,ij)\
r>) x Mi Ln
> m J
=
(n, m # 0 ) ,
J (<^Mm " w ^
( w
w
)^
-Mm _ ^ n ) / ^
[ p , X n ] = X n (X =
[/3„, ajM] = (1 - a T " " ) ^ }
(M + v± 0) + A ; n W ^ ^ + m , o (M + V = 0) ,
(21)
(3,x^),
and the Serre relations. Since these two current basis are basis of the same Lie algebra sljy, they are related by linear transformation, N
N-u E
PNm+v - 22 m
+V
+
2J 208
ii,i+v—N
^™ •'m+l
Comments N-v
»^L
+,
on the Deformed W^ Algebra
2061
N
ii+u 1)E + /
= E^
~ ^ '+
JV-1
k
.00 _ v LUHHjp
E
uf^-vifttr",
(22)
K
where m E Z and 1 < fJ,, u < N— 1. Here, for simplicity of the presentation, we have introduced glN generators E%j (n G Z, 1 < i, j < AT), which satisfy [E^,E1^'] = pi E^m — 6%* E^^ + 6li Sjt kndn+m,0, and the generators in the homogeneous picture are expressed as E+'* = E%i+1, E'^ = E^1'* and H^ = E^ - Eln+1'i+1. We remark E^ = [E]£, El,ltm] (i ¥= J) a n d this RHS is independent on I and m. Next let us consider the splitting of the Cartan part: ( S(JV generator) = (exponential of Cartan generators) x (new generator),
(23)
where new generator commutes with Cartan generators. For homogeneous gradation, the algebra generated by these new generators is known as the (sljv version of) parafermion algebra of level k. For principal gradation, we name it as (sljv-version of) Z algebra of level k, ZA(sIjv)fc- (N = 2 case was studied by Lepowsky and Wilson.3) Explicitly the generators of ZA(slN)k, z£ (n e Z, 1 < fi < N - 1, fi is understood as mod N), are obtained by v( C s y) -= .: ^e nx
p (k - ^ i ( l - ^ " ) / 3 n r n ) :*"(C),
(24)
where : * : stands for the normal ordering for boson (3n and we have introduced currents XM(Q = E n e z ^ C - " a n d ^(0 = E n e z ^ C ~ n - Then (21) implies the relation of ZA(sljv)fc)
^(£)^(CiK(C 2 ) - ^(C 2 )^(Ci)^(|) kD6(cj»&)
(/i + t/ = 0),
where D = £ ^ , ^ ( C ) — Engz n C™ 5 ^(C)
an
d
tne
structure function <7M'"(C) is given by
- exp(-i Y, ^ " wM")(1 " ^ " K " ) -
(26)
n>0 -^0
Next we present an interesting relation between DWA(sljv)q,t and ZA(sljv)fc- Let us consider the following limit c : Limit II : \ ; _\ Lt—u q « , c
k_±si K : fixed.
27
For this choice of t = u; *g w , we cannot take Limit I because /3 = ^ ± ^ — | ^ | depends on H. 209
2062 S. Odake
We assume that DWA currents Wl(z) have the ^.-expansion W^p^C) = hu^z\Q + 0(h2).
(28)
Then, under the Limit II, we can show that the relation of DWA(slN)q,t (10) reduces to that of ZA(sljv)fe (25). (Eq.(10) begins from h2 term and its coefficient is (25). We remark that in this derivation we do not use free field realization at all.) In -—•.
-
k-\-N
other words, DWA(s[jv)g,t with t = UJ q N can be considered as a ^-deformation of ZA(sljv)fc; which we denote as DZA(sljv)fe, DZAfeU)* = DWA(sljv) ^ . (29) q,t=w~1q
N
Concerning the free field realization, however, our assumption (28) does not hold on the Fock space except for N = 2 case. But calculation of some correlation functions supports the assumption (28); We have checked (A|W/1(^i) • • • W1((^n)\X) = 0(hn) for n < 6. We guess that the assumption (28) holds on the level of correlation functions, or, on the irreducible representation space obtained by taking some BRST cohomology. For N = 2 case, (28) holds on the Fock space, and screening currents and vertex operators of DVA (after some modification of zero mode) reduce to those of ZA. Finally we mention the character of DZA(sl2)fc for k € Z>2, i.e. that of DVAg!t = DWA(si 2 ),,t with t = e-niq*¥. We write W1^) and w1(X) as T(() and A respectively, e.g., the highest weight state is defined by Tn\\) = X\\)5no (n > 0). Since degenerate representations of DVA occur at A = Ar,a = t%q~i + i ^ S g i , 1 let us consider A = A: j+i±z (J = — f>— f +1>'"') f ) representations. Grading operator p satisfies [p, Tn] — nTn and —p\X) — (4LX2) ~ I ) W- ^ e character of DZA is defined by xf2A(T) = iiy~p where y = e2ntT and the trace is taken over irreducible DZA spin j representation space. Shiraishi and present author conjectured d XDZA(T)
Here Xr,s'P
=ym~i
1 J ] ( - i ) Y O ^ ) =»^*"*X?j!^(r). (30)
(T) is the Rocha-Caridi character formula 1
•y{p',p")fT\ —
Y ^ /y(p"r-p's+mp'p")m
(2/;y)oomeZV
_
(r+mp')(s+rnp")\
/gj\
/'
which gives the character of the Virasoro minimal representation when p' and p" are coprime, (p',p") = 1. In the above case, p' = 2 and p" = k + 2 imply that ip',p") = 1 for odd k but (p',p") = 2 for even A;. When q is not a root of unity, by studying the Kac determinant of DVA,1 we can check that (30) is true. We remark that the character of DZA(sl2)fc, xfZA> coincides with that of ZA(sl2)fc which is obtained by using the result of Ref. 11, BRST structure of principal sl2d
We remark that this character appears in the calculation of the one-point local height probability of the Kashiwara-Miwa model ( M. Jimbo, T. Miwa and M. Okado, Nucl. Phys. B275[FS17] (1986) 517-545). 210
Comments on the Deformed WM Algebra 2063
Acknowledgements The author would like to thank H. Awata and J. Shiraishi for valuable discussions, and M. Jimbo for helpful comments and explaining poles of f'l and efficient graphical computational method for DWA correlation functions. He is also grateful to M.L. Ge for kind invitation to this APCTP-Nankai joint symposium and thanks members of Nankai Institute for their kind hospitality. This work is supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, No.12640261. A p p e n d i x A . DWA(sIjv) q ,t a n d ^-function regularization ( b y H . A w a t a , S. O d a k e a n d J . Shiraishi) In this appendix we present an interesting relation between DWA(s[jv)g,t and £function regularization. 5 In string theory, 12 the physical state condition is given by (L0 - l)|phys) = 0 (and its antichiral counterpart), where LQ is the zero mode of the Virasoro generator. This condition and the space-time dimension are derived by careful study of string theory (Lorentz invariance in the light-cone gauge, nilpotency of BRST charge, etc.), but there is a shortcut method, ( function regularization method. First we illustrate this method by taking a bosonic string theory as an example. In the light-cone gauge the Virasoro generator Ln is given by Ln = Y%t\ EmeZ \ : aln_mazm : where a%n (n G Z, i — 1, • • •, 24) satisfies [a'n, a3m\ — nSzj6n+mfi and : * : stands for the normal ordering. The Virasoro zero mode without the normal ordering is Lg° N ° = YZx E „ e z K X = E i = ! E n e z \ •• °tn< : +12"E„>o™"- Of course the sum " 53n>o n " 1S divergent and this expression is meaningless. But we replace the sum " X ^ n > o n " by C(—1), where £(z) is the Riemann £ function. Then the above physical state condition is equivalent to the condition that the Virasoro zero mode without the normal ordering annihilates the physical state: LS°N°|phys)=0,
i g o N O = L0 + 12C(-l),
(A.l)
because of the value £(—1) = — ^ . We might say that the Virasoro generator "knows" the value C(~l)Next let us mimic the above procedure for DWA(slN)qj case. DWA current without the normal ordering becomes Wi n o N O (z) = " / M ( l ) _ * " w\z), where " / M ( l ) " is divergent for generic (3 (recall t = q@ and q = eh). Let a\m be coefficients of the following ^-expansion (1 - o 4™ W " Then /••*(*) is /*•*(*) = e x p ( £ „ > 0 £ E r o > o 4m(nh)2m zn). We define (-regularized /c-reg (1) ^ exchanging these summations over n and m and replacing ^2n>0 n 2 m _ 1 with ((1 — 2m) as follows: / ^ e g ( l ) = e x p ( £ 4 m C ( l - 2m)h2m).
(A.2)
m>0
In the Limit I (4), DWA current behaves as Wl{z) 211
= (^) + 0(h2), which can be
2064
S. Odake
shown by using free field realization. 7 So we require t h a t /3 = ^ ^ or j$rj, which corresponds to the vanishing Virasoro central charge, and the zero mode of t h e i - t h DWA current without normal ordering takes the above value ( i ) on t h e v a c u u m s t a t e |vac), which is characterized by ft^|vac) = 0 (n > 0, Vi), W, i noNO |vac) = (^)|vac),
W* n o N O (z) = / ^ e g ( l ) ^ W\z).
(A.3)
Since we can show Wd|vac) = ^ |vac), this requirement implies
^jyj^UjT, [n]! = [n] • • • [1] and [n] = p | ~ p ~ f • We can check that this
where L ' J
L«-J»L-" — - J = " • '
- *
- "
- "
p3-p~2
equation really holds by using formulas log(sinha;) = log a; + Z ^ o C - ! ) " - 1 2n
2
(2n)!n"
m Bm
x (0 < \x\ < IT) and ((1 - 2m) = ( - l ) f2m ^ (m = 1,2,---). Here Bn is the Bernoulli number defined by ^ + f = 1 + £ „ > O ((_1\n-l - 1 ) ' 1 _ 1 ( ! ? 7 T : E 2 " (M < 2*r). (2n)!
Therefore we might say that DWA(sljv) g ,t (with t = g T r ^ K + r ) for e a c h TV "knows" all the values ( ( 1 - 2m) (m = 1,2, • • •)• References 1. See references in the following article : S. Odake, hep-th/9910226. To appear in CRM series in mathematical physics, Springer Verlag. (Errata: http://azusa.shinshuu.ac.jp/~odake/paper.html) 2. Y. Hara, M. Jimbo, H. Konno, S. Odake and J. Shiraishi, Proceedings of the Conference on InGnite-Dimensional Lie Theory and Conformal Field Theory, edited by S. Berman, P. Fendley, Y. Huang, K. Misra, and B. Parshall, CONM series AMS. 3. J. Lepowsky and R.L. Wilson, Proc. Natl. Acad. Sci. USA 78 (1981) 7254-7258. 4. J. Shiraishi, talk at conference "MATHPHYS ODYSSEY—Integrable Models and Beyond—", 19-23 February 2001, Okayama and RIMS Kyoto. 5. This result was obtained by H. Awata, S. Odake and J. Shiraishi, and first announced at Chubu Summer School, 2-5 September 2000, Yamanaka-ko Japan, by S. Odake. 6. B. Feigin and E. Frenkel, Comm. Math. Phys. 178 (1996) 653-678. 7. H. Awata, H. Kubo, S. Odake and J. Shiraishi, Comm. Math. Phys. 179 (1996) 401-416 ; q-alg/9612001. To appear in CRM series in mathematical physics, Springer Verlag. 8. E. Frenkel and N. Reshetikhin, Comm. Math. Phys. 197 (1998) 1-32 ; P. Bouwknegt and K. Pilch, Adv. Theor. Math. Phys. 2 (1998) 125-165. 9. Y. Hara, M. Jimbo, H. Konno, S. Odake, and J. Shiraishi, J. Math. Phys. 4 0 (1999) 3791-3826. 10. See, for example, V.G. Kac, Infinite dimensional Lie algebra, 3rd ed., Cambridge University Press, 1990. 11. Y. Hara, N. Jing and K. Misra, math.QA/0101226. 12. See, for example, M.B. Green, J.H. Schwarz and E. Witten, Superstring theory vol. 1, 2, Cambridge Univ. Press, 1987.
212
Internationa] Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2065-2069 © World Scientific Publishing Company
T H E PEIERLS SUBSTITUTION A N D T H E V A N I S H I N G MAGNETIC FIELD LIMIT
STEPHANE OUVRY Laboratoire de Physique Theorique et Modeles Statistiques, 91405 Orsay, Prance
Bat. 100, Universite
Paris-Sud,
Received 16 October 2001 In order t o achieve in configuration space a dimensional reduction from dimension two to dimension one, the lowest Landau level (LLL) projection, also called the Peierls substitution, is not sufficient. One has also, once in t h e LLL, to take the vanishing magnetic field limit.
It is commonly believed that projecting a bidimensional (2d) 1-body system onto the LLL of an external homogeneous magnetic field makes it essentially unidimensional (Id), due to the dimensional reduction of the 1-body phase space from four to two dimensions. Numerous applications have used this line of reasoning, usually referred to as the Peierls substitution. 1 Starting from a 2d Hamiltonian H = -2dd + uc{zd - zd) + \u\zz
+ V^z, z)
(1)
for a particle in a scalar potential V\(z,z) coupled to a strong magnetic field (we assume without any loss of generality that eB > 0, u>c = +eB/2 is half the cyclotron frequency) and projecting it onto the LLL iKz) = f{z)e-^zz
(2)
where f(z) is analytic, one obtains an eigenvalue equation which, in its modern reformulation,2 rewrites as wc+:V1(z,±d)!)f(z)=Ef(z)
(3)
where the normal ordering :: means that — d is put on the left of z. Clearly, the commutative 2d space has been traded for a non commutative space (Id phase space like)
[-70.,*] = -7
(4)
However, this system is still bidimensional, as can be readily seen on its partition function, which, in the simplest case Vi = 0, scales like the 2d infinite surface of 213
2066
S.
Ouvry
the plane. I will argue that in order to achieve an actual dimensional reduction, i.e. not only in phase space but also in configuration space, the LLL projection is not sufficient per se. One has also, once the system has been projected onto the LLL, to take the vanishing magnetic field limit. 3 Firstly, it might be objected that taking the 5 —> 0 limit in the LLL is counter intuitive: the LLL projection is physically justified when the cyclotron gap is large compared to the temperature and/or the potential (hojc » kT,hwc » V\) so that the excited states above the LLL can be ignored. Thus the LLL projection is associated with a strong B limit, and clearly such an interpretation becomes meaningless when the magnetic field vanishes. However, the algorithm proposed here -LLL projection, then B —> 0 limit-, which basically amounts to ask about the whereabouts of the LLL Hilbert space in the particular limit when its defining parameter, the magnetic field, vanishes, is well defined mathematically. Secondly, the B —¥ 0 limit in the LLL might be a priori ambiguous. Still, it can be given a non ambiguous meaning if the system is regularized at long distance, for instance by a harmonic well of frequency OJ,A and, only after i) projecting onto the LLL harmonic eigenstates -the LLL eigenstates deformed by the harmonic wellii) letting B —> 0, can one take the thermodynamic limit u —> 0. Under these conditions, we will see that a dimensional reduction of the configuration space from dimension two to one is properly achieved. Let us first start by a reminder about what is meant by thermodynamic limit in a 2d harmonic well a : the 1-body spectrum is Enm = (2n + \m\ + l)u
(5)
with n > 0, m positive or negative integer. Clearly, the 2d harmonic well partition function 1 1 —u>->0 (6) u (2sinh^)2-^ (/^; 2 has to be identified in the thermodynamic limit, i.e. when u = 0, with the 2d free partition function Z%=2 — V/(2ir/3), where V is the infinite surface of the 2d plane. Therefore the 2d thermodynamic limit prescription should be that when UJ —> 0
Let us now consider, in the presence of a magnetic field, the Landau Hamiltonian here for convenience expressed in the symmetric gaugeb HL = -2dd
+ wc(zd - zd) + -OJ2CZZ
(8)
a T h e first author to use a harmonic well regularization was E. Fermi, see cond-mat 9912229 where the original article of Fermi 5 is translated. b Considering rather the asymmetric gauge where the Landau eigenstates are product of a plane wave on one axis and a Hermite polynomial on t h e other axis would not help t o understand the dimensional reduction mechanism.
214
The Peierls Substitution
and the Vanishing Magnetic Field Limit
2067
and its spectrum Enm = (2n + \m\ + l)uic - muc
(9)
with an infinite degeneracy per Landau level eBV/(2n) - the product of the infinite surface of the 2d plane by the magnetic field strength. In the LLL, n = 0, m > 0, and E = uc, the 1-body LLL eigenstates are analytic (up to the Landau gaussian factor) . .m+l
C^—r )izme-^'zZ
(10)
As already said, the LLL partition function ZLLL = ^ e " * -
(11)
is obviously 2d since it scales like the surface V of the 2d plane. What happens in the limit B —»• 0? Here an ambiguity arises due to the vanishing field strength multiplying the infinite surface of the plane. In order to cure this ambiguity, let us confine the system in a harmonic well, so that the 1-body LLL harmonic eigenstates (i.e the deformation of the LLL eigenstates n = 0, m > 0 by the harmonic well) are still analytic (up to the Landau-harmonic gaussian factor) {-L-T)\zme~^zz
(12)
but now the spectrum is non degenerate E = u)t + (ojt - uc)m
(13)
with cjt = \jw*. + u>2. The LLL harmonic partition function becomes Z
LLL+U,
=
1
_ e_0(ut.UB)
(14)
At this point, one should keep u) fixed, let B —»• 0, and then take the thermodynamic limit u> —> 0. Before doing so, let us check that by keeping B fixed but taking the thermodynamic limit u) —> 0, one correctly recovers ZLLL- Since, when CJ —> 0, w* — u)c ~ UJ2l{2u)c) one indeed gets, using the 2d thermodynamic limit prescription (7), e-/3o;e
ZLLL+U
— -57
eQ
r ->w=o —-Ve~^c
= ZLLL
(15)
Note that what we found here is yet an other way to actually show that the Landau degeneracy is, in the thermodynamic limit, eBV/(2ir). Now in the case of interest, first set B — 0, i.e. u)t = w, then take the thermodynamic limit u -> 0, one gets, still using (7), *LLL+u> =
1 / " . ~ - 3 - ->u;=0 \\ T^a 0W 2 s i n h I^Y w->o P<^
215
vd=i o
= Z
(16)
2068
S.
Ouvry
i.e. the Id partition function for a free particle on a line of infinite length L = y/V. At the level of the spectrum, being in the LLL and a harmonic well, and taking, as advocated above, the B —> 0 limit, the LLL harmonic basis (12) and spectrum (13) have narrowed down to m+l
( irm
m
r)*z
,
m>0
(17)
and w(m + l),
m>0
(18)
This amounts to pick up on each 2d harmonic energy level (j + 1)UJ, j > 0, with degeneracy j +1, the eigenstate of maximal angular momentum j , and consequently zero radial quantum number, yielding the spectrum (18) which happens to coincide with a Id harmonic spectrum, provided that the 2d positive angular momentum quantum number m is now interpreted as the Id harmonic quantum number. It is therefore manifest, both on the partition function and on the spectrum, that a dimensional reduction from d=2 to d = l has been achieved. To put it bluntly, we have shown that
and accordingly for the density of states PLLL(E)
= ^V5(E
-
Wc )
-»• pi=\E)
= - ^ L
(20)
where PLLL(E) and pf=1(E) stand respectively for the LLL and the free Id density of states. Clearly, setting directly B = 0 in ZLLL or in PLLL{E) has no meaning whatsoever. Still, (19,20) have been given a non ambiguous meaning through the long distance harmonic regularization. Up to now one has dealt with spectra and partition functions. The same logic applies as well to the Hamiltonian and the eigenstates: consider (1) but now in a harmonic well H = -2dd + u)c{zB - zd) + -u%zz + Vi(z, z)
(21)
and project it onto the LLL-harmonic basis i>{z) = f(z)e-^zs
(22)
to obtain the eigenvalue equation6 Lt
+ {ut - ujc)zd+ : V^z, - U ) :) f(z) =
Ef(z)
(23)
When B —• 0 it becomes u(l
+ zd+ : V^z, i d ) :) f(z) = 216
Ef(z)
(24)
The Peierls Substitution and the Vanishing Magnetic Field Limit
2069
acting on ijj{z) = f(z)e~*zz. It is obvious that the kinetic p a r t of t h e Hamiltonian (24) is nothing but the I d harmonic well Hamiltonian in a coherent s t a t e representation, with the mapping zm -> Hm(x) between t h e 2d a n a l y t i c function zm and the I d Hermite polynomial Hm(x). Moreover, looking at t h e potential Vi, one realizes that the 2d commuting space has been traded for a n o n c o m m u t a t i v e space
In the thermodynamic limit, w —> 0, one gets an infinite n o n c o m m u t a t i v i t y which should be viewed as the signature of the dimensional r e d u c t i o n which has t a k e n place from a 2d to a I d system.
References 1. R. Peierls, Z. Phys. 80, 763 (1933) 2. S. Girvin and T. Jachs, Phys. Rev. D29, 5617 (1984); G. Dunne and R. Jackiw, hepth/9204057 3. S. Ouvry, cond-mat 9907239, Phys. Lett. B510, 335 (2001), where, as an illustration, the equivalence of the 2d LLL-anyon model and the Id Calogero model was established in the vanishing magnetic field limit. 4. A. Comtet, Y. Georgelin and S. Ouvry, J. Phys. A: Math. Gen. 22, 3917 (1989); K. Olaussen, "On the harmonic oscillator regularization of partition function", Trondheim Univ. preprint No. 13 (1992), cond-mat/9207005 5. "Sulla quantizzazione del gas perfetto momoatomico", E. Fermi, Rend. Lincei 3, 145 (1926) 6. for a general description of the Peierls substitution in the vanishing magnetic field limit, its generalization to higher Landau levels, and their reformulation in terms of non commutative geometry, see N. Macris and S. Ouvry, in preparation.
217
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2071-2077 © World Scientific Publishing Company
APPLICATIONS OF A H A R D - C O R E B O S E - H U B B A R D MODEL TO WELL-DEFORMED NUCLEI*
YUYAN CHEN, FENG PAN Liaoning Normal University, Department of Physics, Dalian 116029, P. R. China G. S. STOITCHEVA, and J. P. DRAAYER Department
of Physics & Astronomy, Louisiana State Baton Rouge, LA 70803, USA
University,
Received 29 October 2001 An exactly solvable hard-core Bose-Hubbard model, which is equivalent to a meanfiled plus nearest-level pairing theory, for a description of well-deformed nuclei is used and applied t o t h e actinide region. Binding energies and pairing excitation energies of 226-234 T h i 230-240 T j i a n d 236-243p u i s o t o p e s a r e calculated and compared with the corresponding experimental values.
1. I n t r o d u c t i o n Pairing is an important residual interaction in nuclear physics. Typically, after adopting a mean-field approach, the pairing interaction is treated approximately using either Bardeen-Cooper-Schrieffer (BCS) or Hartree-Fock-Bogolyubov (HFB) methods, sometimes in conjunction with correction terms evaluated within the Random-Phase Approximation (RPA). However, both BCS and HFB approximations suffer from serious difficulties, the nonconservation of the number of particles being one that can lead to serious problems, such as spurious states, nonorthogonal solutions, etc. Another problem with these approximations is related to the fact that both BCS and the HFB methods break down for an important class of physical situations. A remedy in terms of particle number projection complicates the algorithms considerably, often without yielding a better description of higher-lying excited states that are a natural part of the spectrum of the pairing Hamiltonian. Over the past few years progress has been made in the development of better algorithms that bypass the Bogolyubov transformation and thus are free of problems related to particle number nonconservation.1,2 In these approximation, either a configuration-energy truncation scheme or a many-body Fock-space basis cutoff was used, so the results were still not exact. 'Dedicated to Professor Wu F. Y. on his 70th Birthday Celebration 219
2072
Y. Chen et al.
Exact solutions of the mean-filed plus pairing model were first studied for the equal strength pairing model. 3 - 5 Recently, generalizations that include state dependent pairing have been considered. 6-9 In these cases, the Bethe ansatz was used, from which excitation energies and the corresponding wavefunctions can be determined through a set of nonlinear equations. Unfortunately, solving these nonlinear equations is not practical when the number of levels and valence nucleon pairs are large, which is usually the case for well-deformed nuclei. 2. A Hard-core Bose-Hubbard Model for nuclei In Ref. 9, a hard-core Bose-Hubbard model was proposed, which is equivalent to a mean-field plus nearest-level pairing theory. As is well known, an equal strength pairing interaction, which is used in many applications, is not a particularly good approximation for well-deformed nuclei. In Ref. 2, a level-dependent Gaussian-type pairing interaction with dj = Ae-B^-^)2
(1)
was used, where i and j each represent doubly occupied levels with single-particle energies e^ and tj. The parameters A < 0 and B > 0 are adjusted in such a way that the location of the first excited eigen-solution lies approximately at the same energy as for the constant pairing case. Of course, there is some freedom in adjusting the parameters, allowing one to control in a phenomenological way the interaction among the levels. Expression (1) implies that scattering between particle pairs occupying levels with single-particle energies that lie close are favored; scattering between particle pairs in levels with distant single-particle energies are unfavored. As an approximation, this pairing interaction was further simplified to nearest-level coupling in Ref. 9, namely, Gj, is given by (1) if the levels i and j lie adjacent to one another in energy, with Gy taken to be 0 otherwise. Hence, the Hamiltonian can be expressed as
where the first sum runs over the orbits occupied by a single fermion which occurs in the description of odd-A nuclei or broken pair cases, and the second primed sum runs only over levels that are occupied by pairs of fermions. For the nearest-level pairing interaction case the i-matrix is given by ta = 2ej + Ga = 2e» + A and Ui+i — U+u = Gu+i with tij = 0 otherwise. The fermion pair operators in this expression are given by
h+ = ai+aj+, 220
bi = ajOi,
(3)
Applications
of A Hard-Core Bose-Hubbard Model
2073
where Oj + is the i-th level single-fermion creation operator and at the corresponding time-reversed state. The bi+ and bi satisfy the following commutation relation: [bi,bj+] = 6ij(l-2Nj,
[Ni,bj+]=6ijbj+,
[Ni,bj] = -6ijbj
(4)
where Ni — \(a,i+cn + a-^aj) is the pair number operator in the i-th level for even-even nuclei. In this paper the Nilsson Hamiltonian is used to generate the mean-field. In this case there is at most one valence nucleon pair or a single valence nucleon in each level due to the Pauli principle. Equivalently, these pairs can be treated as bosons with projection onto the subspace with no doubly occupied levels.9 The eigenstates of (2) for fc-pair excitation can be expressed as = Yl'il
|fc; £, (njl, nj2, • • •, njr)nf) hjbij
• • • bij\(nh,nh,
*
• • ^n^rif),
(5)
where ji,J2, • • •, j r are the levels occupied by r single particles, the prime indicates that ii, 12, •• •, ik can not be taken to be ji, J2, •••, jr in the summation, and n; is the total numbers of single valence nucleons, that is rif = J2j nr Since only even-even and odd-A nuclei are treated without including broken pair cases in this paper, r is taken to be 1 for odd-A nuclei, and 0 for even-even nuclei. In Eq. (5), ^hL—ik i s a determinant given by 9^ 2
gj
of"
9^ 2
gf
qf
k
•••
9*1
•••
gf2
(6)
... qf"
where £ is a shorthand notation for a selected set of k eigenvalues of the t matrix without the corresponding r rows and columns denoted as t, which can be used to distinguish the eigenstates with the same number of pairs, k, and g^p is the p-th eigenvector of the t matrix. The excitation energies corresponding to (5) can be expressed as
where the first sum runs over r Nilsson levels each occupied by a single valence nucleon, which occurs in odd-A nuclei or in broken pair cases, the second one is a 221
2074
Y. Chen et al.
sum of k different eigenvalues of the i-matrix. Obviously, i is a (k — r) x (k — r) matrix, since those orbits occupied by single valence nucleons are excluded resulting from the Pauli blocking. #(&•) is the p-th. eigenvalue of the i-matrix, that is
Y^kg^^E^gt*.
(8)
3
Hence k
H\k;Z,(njl,nh,---,njr)nf)=
£
£ $ > )
h
r P
£ >
M=l P
= £fc° I*! £. ("JI . "ja ' • • n3k )»/),
;
+
E^^)x
*=1
(9)
where P runs over all permutations, E^^ is the /i-th eigenvalue of the i matrix. Eq. (9) is valid for any k. If one assumes that the total number of orbits is N for even-even nuclei, the fc-pair excitation energies are determined by the sum of k different eigenvalues chosen from the N eigenvalues of the i matrix with r — 0, the total number of excited levels is N\/kl(N — k)\. While for odd-A nuclei or broken pair cases, the levels that are occupied by the single valence nucleons should be excluded in the original t matrix. In the latter case, the eigenvalue problem (4) can be solved simply by diagonalizing the corresponding i matrix as shown in Eq. (9). 3. Applications to Actinide Isotopes In this section, we try to describe nuclei in the actinide region with the mean-field plus nearest-level pairing model using the axial-symmetric Nilsson potential as the mean-field. Other than what is manifest through the mean field, the quadrupolequadrupole interaction is not considered. In this case, exact solutions can be obtained by using the above simple method. As for the binding energy, the contributions from real quadrupole-quadrupole interaction is expected to be relatively small. 10 This conclusion applies to low-lying 0 + excited states as well as ground states. As shown in Ref. 11, contributions from the pairing interaction is very important to the low-lying excited 0 + states in these deformed regions. Hence, the position of low-lying 0 + states is an estimate based on the Nilsson mean field plus pairing approximation. In this well-deformed region there are a lot of nuclei. The parameters were fixed by considering the 226-234Th; 230-240TJ) a n d 236-243pu i s o t o p e s . Specifically, the binding energies of these isotopes were calculated. Table 1 shows the binding energy results as well as pairing excitation energies of the theory for 226-234']^ 230-240TJ 5 222
Applications of A Hard-Core Bose-Hubbard Model 2075
and 236-243pu^ -wifa tjj e corresponding experimental values taken from Ref. 12. The parameters A and B in Eq. (1) were fit as follows to maximize agreement with experiment: A = QI +/3ifc + 7ira/, B = a2 + fok +
(10)
^nf,
where a,, Pi, and 7, are parameters that were fit for each isotope.
Table 1. Calculated binding and pairing excitation energies are compared with the corresponding experimental values for various 2 2 6 - 2 3 4 Th, 2 3 °- 2 4 0 U, and 2 3 6 - 2 4 3 p u isotopes. Bth(MeV) and Sexp(MeV) denote, respectively, the theoretical and experimental binding energies.
Nucleus
226 rTYL
Spin and Parity 0+
Bexp(MeV)
-1730.54
Btfc(MeV)
Pairing excitation Energies of Exp. (MeV)
-1732.17
02+ 1 + 22^ 1 + 23 1 +
227Th
1+ 2
-1736.00
-1733.97
228rpi
0+
-1743.10
-1739.30
229Th
5 + 2
-1748.36 -1755.16
-1744.42 -1756.90
24
230rp.
231Th
0+ 5+ 2
-1760.27
-1764.21
o5 2 + 22^ 5 + 23 o2+ 5 + 22^ 5 + 23 5 + 24^
0+ 1+ 2
-1766.71 -1771.50
234Th
0+
-1777.69
-1779.81
231
U
52
232TJ
0+
-1758.72 -1760.00
-1761.26 -1758.94
02
233 , J
5+ 2
-1770.23
5 + 22^ 5 + 23^
234
u
0+
235yj
72
-1771.74 -1778.59
-1774.41
-1780.23
223
0.317 0.730 1.079 0.310 0.810 1.150 1.470
1 + 22
o2++ 03 04+
— +
02+ 03+
o4+
-1783.89
0.317 0.635 0.241 0.302
02+ 03+
233o-iu
7 22 7 23
o2+
1 +
2
5.188 6.495 0.831 0.029
,+
232 rpi
-1768.66 -1772.92
0.805 3.226
Pairing excitation Energies of Th. (MeV)
2
L_
1 + 23 1 + +
1.415 0.718 0.057
02+
0.516 1.199 0.907
54
,
02 5 + 22j_ 5 + 23
5 + 22^ 5 + 23 5 + 24 L
02+ 03+
1 + 22 +
02
o3++ o5 3 22
0.691 0.340 0.546 0.809 1.044 1.781 0.670 0.700
0.999 1.299 1.391
o2+
5 + 22 L 5 + 23^
o2++ o3+ o4
7 22 7 23
1.204 1.230 1.647 2.585 0.907 1.066 2.562 2.904 0.646 0.961 0.732 0.803 0.747 0.933 1.696 0.826 1.056
2076
Y. Chen et al.
Table 1 (Continued)
Nucleus
236
U
237
U
238
U
Spin and Parity
0+
Bexp(MeV)
-1790.44
1+ 2
-1795.56
0+
-1801.715
Bt/j(MeV)
-1786.71
-1795.48 -1802.22
239
U
5+ 2
-1806.52
-1810.23
240 U 236
0+ 0+
-1812.45 -1790.46
-1815.41 -1792.36
Pu
237
Pu
238
Pu
7~ 3
0+
-1795.56
-1801.72
-1795.87
-1799.96
Pairing excitation Energies of Exp. (MeV) 02+ 03+ 04+ l + 32, 1 + 33
02+ 03+
5 + 32 5 + 33 5 + 24
02+
7 32 7 23 02+ 03+ 04+
239
1+ 3
-1806.52
-1805.12
05 + 1 + 22
240
0+
-1812.45
-1810.68
03+
Pu Pu
o2+
o4
241 Pu 242
Pu
243
Pu
5+ 2
0+ 7 2
-1816.64 -1822.41 -1826.63
-1816.09 -1821.89 -1828.63
+
5 + 32, 5 + 23^
02+
7 32 7 33 7 ~ 24
0.919 2.155 2.750 0.846 0.905 0.925 0.993 0.193 0.734 0.757 3.000 0.691 0.696 0.942 1.134 1.229 1.427 0.753 0.860 1.089 1.526 0.233 0.801 0.956 0.333 0.450 0.742
Pairing excitation Energies of Th. (MeV) 02+ 03+ 04+ l + 32 1 + 33, 02 +
03+ 5 + 32 5 + 23 5 + 24 ,
o2++ o2
7 32 7 23 o2+
o3+ 04 +
05+ 1 + 32 +
02
o3+ 04 + 5 + 32^ 5 + 33
02
+
7 32 7 23 7 24
0.913 1.186 2.319 0.586 0.700 0.877 2.874 0.185 0.459 0.786 0.100 0.645
0.617 2.173 0.407 1.987 2.170 2.681 0.354 1.030 2.144 2.626 0.088 0.587 1.186 0.845 1.146 1.815
Acknowledgements This work was supported by the U.S. National Science Foundation through a regular grant (9970769) and a Cooperative Agreement (9720652) that includes matching from the Louisiana Board of Regents Support Fund, and by the Natural Science Foundation of China (Grant No. 10175031) as well as by the Science Foundation of 224
Applications of A Hard-Core Bose-Hubbard Model 2077 the Liaoning Education Commission ( G r a n t No. 990311011).
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
J. Y. Zeng and T. S. Cheng.JVud. Phys. A405, 1 (1981) H. Molique and J. Dudek, Phys. Rev.C56, 1795(1997) R. W. Richardson, Phys. Lett. 5, 82(1963) R. W. Richardson, J. Math. Phys. 6, 1034(1965) R. W. Richardson and N. Shermen, Nucl. Phys. 52, 221(1964) F. Pan and J. P. Draayer, Phys. Lett. B 4 2 2 , 7(1998) F. Pan and J. P. Draayer, Ann. Phys. (NY) 271, 120(1999) F. Pan, J. P. Draayer, and L. Guo, J. Phys. A 3 3 , 1597(2000) F. Pan and J. P. Draayer, J. Phys. A 3 3 , 9095(2000) S. G. Nilsson and O. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. 32, 1(1961) P. E. Garret, J.Phys. G27, Rl(2001) P. Mler, J. R. Nix, W. D. Myers, and W. J. Swiatecki, Atomic Data Nucl. Data Tables 59, 185-381(1995)
225
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2079-2087 © World Scientific Publishing Company
ELLIPTIC A L G E B R A A N D I N T E G R A B L E M O D E L S FOR SOLITONS O N NONCOMMUTATIVE T O R U S T
BO-YU HOU* DAN-TAO PENGt Institute
of Modern Physics, Northwest University Xi'an, Shaanxi, 710069, China
Received 13 November 2001 We study the algebra An, the basis of the Hilbert space Hn in terms of 6 functions of the positions of n solitons. Then we embed the Heisenberg group as t h e quantum operator factors in the representation of the transfer matrices of various integrable models. Finally we generalize our result to the generic 9 case. PACS: 11.90.H-t, ll.25.-w Keywords: noncommutative torus, integrable elliptic models.
1. Solitons on noncommutative plane In the noncommutative plane i? 2 , the coordinates x1 and x2 satisfy the following relation: [xi,x*]=i9,
(1)
here 6 is a constant. The algebra A associated with this space is generated by the functions of a;1 and x2. The functional form of the algebra A is defined by the Moyal * product / * g{x) = eieije^^J
f(x)g(y)\x=y.
(2)
The derivative di is the infinitesimal translation automorphism of the algebra A: xi —• xi + e\
(3)
where el is a c-number. For algebra A this automorphism is internal: dif{x) = i6eij[x'*, f{x)\ = i0ij[x>*, f(x)], here 6a = 9ea * Email:[email protected] tEmail:[email protected] 227
(4)
2080 B.-Y. Hou & D.-T. Peng
The operator form of A is generated by Weyl-Moyal transformation. a* = —7=(x1 + ix2),
a=—={xx-ix2),
(5)
which obey [a,at] = l.
(6)
Since a and a) satisfy the commutation relations of the creation and annihilation operators, we can identify the function / ( x 1 , x2) as the functions of a and a? acting on the standard Fock space % of the creation and annihilation operators: « = {|0>,|1), • • • » , • • • } .
(7)
where |0) and \n) satisfy: (aUn
a\0) = 0,
\n) = ^-p^lO), a)a\n) = n\n). (8) Vn! The Weyl-Moyal transformation maps the ordinary commutative functions onto operators in the Fock space %: f(x) = f[z = x1 — ix2, z = x1 + ix2) —> / ( a , a t ) = J l0y.f{xy\P^a-z)+p{^e^--z)]^
(g)
where:
,
P=y-^.
do)
/—•/,
a—>g,
(ii)
P=^ It is easy to see that if
then f*g-+fg
(12)
and ' d2xf{x)
—> nOTrf(a, a"1).
(13)
/ •
The translations of R2 are generated by di which are isomorphism to A while applying on the Fock space H: di i—> iQijij.
(14)
In paper 4, Harvey, Kraus and Larsen introduced a quasi-unitary operator to generate various soliton solutions in noncommutative geometry. In noncommutative plane R2, this operator is defined as T
^^Tvain, 228
(is)
Elliptic Algebra and Integrable Models 2081
Acting this operator T on the basis of the Hilbert space H, we have T\n) = \n + 1),
(n\T* = {n + l\.
(16)
and r | n ) ( n | T t = |n + l)(n + l|.
(17)
TPnT* = Pn+1,
(18)
This means that
where Pn = \n)(n\ denote the projection operator onto the n-th states and P% = P. Thus we have TT*\n) = \n), (n < 1) and TT*\0) = 0,
(19)
and TT* = 1 — |0> (0| = 1 - P 0 .
(20)
T is the quasi-unitary soliton generating operator. 2. Solitons o n noncommutative torus T and Heisenberg group In the noncommutative torus T, the algebra A is generated by the Wilson loop Ui, (i = 1,2). The arbitrary element a £ A is
a = Y,c,ijM1Ui' For the periodicities / and
2-KIT of
£/i = eilx\
(21)
the torus, the generators of the algebra A are
U2 = ea{-T*xl-Tlx2).
(22)
Since [a;1, a:2] = 10 locally, so UiU2 = U2Uieil2T2e.
(23)
Now let us consider the integral torus case l-^- = A G N (or Z+) i. e. the normalized area A of the torus is an integer. Then the Wilson loop U\ and U2 are commutative Z7iC/2 = U2UL
(24)
We orbifold T into ~— = Tn by introducing Wi = (Ui)±,
(25)
then on Hn, the Hilbert space on Tn, we will have noncommutative algebra An generated by WiW2 = W2Wie^ 229
= W2WHJ
(26)
2082 B.-Y. Hon & D.-T. Peng which satisfy (27)
W? = W? = 1 ,
2-jri
where w = e » . The Basis vectors of the Hilbert space %n are n
Va = £-F-a,fc, (a = 1,2, • • ' , " ) , 6=1 n
n
1
(28) fe=l
j=i
here a = ( a i , 02) & Znx Zn, and i + 21 CTQ(Z)
=
L2^
(29)
(z,r).
5
n
The 9 function can be transformed to a operator form by the Weyl-Moyal transformation: 6»(*) = $ >iirm
T-\-2nimz
6{z) = Y^e™m2T ••U™U%
(30)
Since Wi : U?W? := uj±m : U^U^ :
(31)
we have WtVaizi, ...,zn)=(]l
W2Va{zi,
Ti*> W 2 T V 0 ( * i ,
• • • , Zn) = ( f[ T^Alf^Vaiz!, M=l '
(32)
•••,zn)
(33)
• • • , Zn),
where T^f(z)=f(z1,---,zi
+
(34)
a,.-.,zn).
Substituting the expressions of Va we get WiVa(zi, •••,zn)
= 14+1(21, • • •, zn),
W2Va(Zl, — ,zn) = e2^Va(Zl,
(35)
• • •, z^.
Then the algebra An = {Wa = Waia2 = W?1 W%2}
(36)
is realized as the 2™ x 2™ Heisenberg matrices Ia, (Ia) ab —
(37)
0a-^aita2UJ
2
l%
Corresponding to the di on R , we have a su n (7^t) acting on V.n sun(Tn) : {Ea\a ^ (0,0)}. 230
(38)
Elliptic Algebra and Integrable Models
2083
Here
Ea = {-!)** va(0)Y,U 3
I
£
<*a(Zji)
n ff
^ a{Zji)
k^i
a = ( a i , a 2 ) ^ (0,0) =
(n,n),
-dj
(39)
and
E0 =
-J^dj,
(40)
where Zjk — Zj — Zk, dj = -^-. The commutation relation between Ea and E7 is [Ea, £ 7 ] = (oj~a^
- cu-a^)Ea+J,
or in more common basis, let Eij = ^ZaM{Ia)ijEa,
(41) we have
[Ejk, Eim\ = Ejm5ki — EikSjm.
(42)
This commutation rule can also be obtained from the quasiclassical limit of the representation of the Sklyanin algebra. 19 Since the Wilson loops W\ and W2 acting on the noncommutative covering torus T is to shift Zi to (zi + ^ — 5inr) and (ZJ + ^ — Sin) respectively, we can get the automorphism of E@ E s«2(T) by noncommutative gauge transformation wa G A WxE^Wi1
=uj-a*Ea(Zi), 1
ai
W2Ea(zi)W2; =u Ea(zi).
(43) (44)
Let Ea £ g to act on Va, we find that EaVa = ^ ( I a ) 6 a K -
(45)
WaVa = ^(/ Q )6aV f e ,
(46)
Next, we know that
so on ~Hn, we establish the isomorphism: sun(T)
^^A;
Ea*—>Wa.
(47)
The operator form of the projection operators becomes
-Y/WO(}(I0)ii
= Pi = \Vi)(Vi\
(48)
and the ABS operators is simply £lO^Wl=£|Va+l>(K|
231
(49)
2084
B.-Y. Hou & D.-T. Peng
3. T h e integrable models for the solitons on noncommutative torus T In this section, we will embed the sun(T) derivative operators as the "quantum" operator factors in the representation of the transfer matrix (Lax operator) of the various integrable models i.e. The elliptic Gaudin model on noncommutative space20 is denned by the transfer matrix (quantum Lax operator): £«(«)=
E
wa{u)Ea{Ia)ij
(50)
a^(0,0)
where wa(u) = V fo) an( ^ ^a anc * ^a a r e * n e generators of su(n) (or An-i Weyl) and Gu(n) respectively. This transfer matrix can also be obtained as the nonrelativistic limit of the Ruijsenaars-Macdonald operators. The common eigenfunctions and eigenvalues of Gaudin model is solved in terms of the Bethe ansatz. 21 Now we substitute the difference representation of su(n) Ea (39) into (50), we get a factorized L of the Gaudin model LG(u)ij=E0+ Y, EM) a^(0,0)
= j>( u ,2)i
(5i)
k
where the factors are the vertex face intertwiner
4>(u,z)) = e
i _ 2 1 2
(it + nzj - y~] zk + — — , n r ) .
(52)
l
k
For the Gaudin model on noncommutative torus, the Zi is the origin (position) of the i-th soliton, di as its infinitesimal translation is equivalently to [z^,]. Next, the elliptic Calogero-Moser model is defined by the Hamiltonian: n i=l
i^j
2
where p(z) — d a(z). The corresponding Lax operator is
W„).
_ ta - I £ ,„MzWi _ L,m
_ s-fJ^iA
(54,
This Lax operator can be gauge transformed into the factorized L (51) of the Gaudin model by the following matrix:
G(
s
6{u\ zY,
"^ n^Tfe)
(55)
The C.M. model gives the dynamics of a long distance interaction between nbodies located at Zi (i — 1, • • •, n). On noncommutative torus, it gives the dynamics of n solitons and Zi becomes the position of the center of the i-th soliton. According 232
Elliptic Algebra and Integrable Models
2085
to Ref. 9, the interaction between n-solitons is the Laplacian of a Kahler potential K, which is the logarithm of a Vandermonde determinant. Actually we have
E »& 3) = E d ' lo s I I
i
j^k
(56)
i
and eK(u,z) =
Y^u{zj-zk)(j{nu+r^)
= det(^) = a(nu+1^±)l[a(zi-zj).(57)
The variable u of the marked torus is the spectral parameter or evaluation parameter of Lax matrix Kj. This Ruijsenaars operators are related to the quantum Dunkle operators and the (/-deformed Kniznik-Zamolodchikov-Bernard equations. The eigenfunctions could be also expressed in terms of double Bloch wave as the algebraic geometric methods.22 We will show this in the more familiar formalism of the elliptic quantum group. 4. The Zn X Zn Heisenberg group in case of the general 0 For the generic 0 case, as in paper 23 we find that 9T = j], here 77 is the crossing parameter and the Zn x Zn Heisenberg group of shift of solitons is realized by the Sklyanin algebra STtV. The noncommutative algebra A is realized as Elliptic quantum group ETtTl. The evaluation module of ETyT] is expressed by the Boltzmann weight of the IRF model. n R(u, A) = E *=1
E
i,i ® Ei,i + E " ( " ' A'J)-Ei,t ® Eo\o + E ^ i¥=j i¥"3
U
'
X
v)EiJ
® E3*
( 58 )
where a{u A)
/3(u A) =
> = e(u-v)6(\y
'
e(u - nwx)
(59)
It satisfies the dynamical YBE: R(Ul,u2,
A - r)h^)12R(uuX)13R(u2,
= R(u2, A)23J2(ui, A - 7]h^)nR(Ul
A-
r]hw)23
- « 2 , A) 12
(60)
where R(u, A — r]h^)12 acts on a tensor v\ (g> v2 <8> ^3 as R(u, A — 77/x) ® Id if U3 has weight /i. The elliptic quantum group ET^{sln) is an algebra generated by a meromorphic function of a variable h and a matrix L(z, A) with noncommutative entries: R(Ul - u 2 , A - f]h^)uL(Ul,
X)13L(u2, A -
= L(u2, X)23L(Ul, X - 7]h^)13R{Ul 233
rjh^)23
- u2, A) 12 .
(61)
2086
B.-Y. Hou & D.-T. Peng
here L(z, A) gives a n evaluation representation of the quantum group
T h e Transfer m a t r i x of I R F is expressed by the Ruijsenaars operators which gives t h e d y n a m i c s of solitons N
T(u)f(X) = J2 Lu{u, A)/(A - Vh)
(63)
a n d t h e R u i j s e n a a r s - M a c d o n a l d o p e r a t o r M is
. ."7.
"
^
i])
So we h a v e Tif(X)
= f(\i - r,b)
(65)
T h e n t h e Hilbert space of n o n - c o m m u t a t i v e torus becomes the common eigenvectors of t h e transfer m a t r i x . T h e wave functions have the form Tp^Yle^Heizi+U-r})
(66)
i
which will b e twisted by r\ when Zi changed by Wilson loop U\, U2-
References 1. J . A . Harvey, Komaba Lectures on Noncommutative Solitons and D-branes, hep-th/0102076. 2. M. R. Douglas, N. A. Nekrasov, Noncommutative Field Theory, hep-th/0106048. 3. R. Gopakumar, S. Minwalla, A. Strominger, hep-th/0003160, JHEP 0005 (2000) 020. 4. J. A. Harvey, P. Kraus, F. Larsen, h e p - t h / 0 0 1 0 0 6 0 , JHEP 0012 (2000) 024. 5. E. Witten, Noncommutative Tachyons and String Field Theory, hep-th/0006071. 6. I. Bars, H. Kajiura, Y. Matsuo, T. Takayanagi, Phys. Rev. D63 (2001)086001, hep-th/0010101. 7. E. M. Sahraoui, E. H. Saidi, Class. Quant. Grav. 18 (2001) 3339, hep-th/0012259. 8. E. J. Martinec, G. Moore, Noncommutative Solitons on Orbifolds, h e p - t h / 0 1 0 1 1 9 9 . 9. R. Gopakumar, M. Headrick, M. Spradlin, On Noncommutative Multi-solitons, hep-th/0103256. 10. T. Krajewski, M. Schnabl, h e p - t h / 0 1 0 4 0 9 0 , JHEP 0108 (2001) 002. 11. H. Kajiura, Y. Matsuo, T. Takavanagi, hep-th/0104143, JHEP 0106 (2001) 041. 12. A. Connes, M. R. Douglas, A. Schwarz, hep-th/9711162, JHEP 9802 (1998) 003. 13. N. Seiberg, E. Witten, h e p - t h / 9 9 0 8 1 4 2 , JHEP 9909 (1999) 032. 14. M. Rieffel, Pacific J. Math. 93 (1981) 415. 15. L. Debrowski, T. Krajerski, G. Landi, Int. J. Mod. Phys. B 14 (2000) 2367. 16. H. Bacry, A. Crossman, J. Zak, Phys. Rev. B 12 (1975) 1118. 234
Elliptic Algebra and Integrable Models 2087 17. J. Zak, in Solid State Physics, edited by H. Ehrenreich, F. Seitz and D. TurnbuU, (Academic, New York, 1972), Vol. 27. 18. K. Chen, H. Fan, B. Y. Hou, K. J. Shi, W. L. Yang, R. H. Yue, Prog. Theor. Phys. Supp. 135(1999)149. 19. E. K. Sklyanin, Funct. Anal. Appl. 16(1982)263, ibid. 17(1983)273. 20. E. K. Sklyanin, T. Takebe, Phys. Lett. A 219(1996)217. 21. B. Feigin, E. Renkel, N. Reshetikhin, Comm. Math. Phys. 166(1994)27. 22. I. Krichever, Elliptic Solutions to Difference Non-linear Equations and Nested Bethe Ansatz Equations, s o l v - i n t / 9 8 0 4 0 1 6 . 23. B. Y. Hou, D. T. Peng, K. J. Shi, R. H. Yue, Solitons on Noncommutative Torus as Elliptic Algebras and Elliptic Models, h e p - t h / 0 1 1 0 1 2 2 .
235
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2089-2095 © World Scientific Publishing Company
N E W RESULTS FOR SUSCEPTIBILITIES IN P L A N A R ISING MODELS*
HELEN AU-YANG and JACQUES H.H. P E R R t Physics Department,
Oklahoma State University, Stillwater,
OK 74078,
U.S.A.
Received 1 January 2002 We briefly review recent progress on calculating susceptibilities in planar Ising models.
1. Introduction First of all, a project like the current one cannot be undertaken by a single person. We owe a lot to our collaborators, teachers, and colleagues, especially R.J. Baxter, H.W. Capel, A.J. Guttmann, M. Jimbo, B.-Q. Jin, X.-P. Kong, T. Miwa, B.M. McCoy, B.G. Nickel, W.P. Orrick, M. Sato, and T.T. Wu. The literature on the two-dimensional Ising model also is very extended. Therefore, we shall only give limited citations, and encourage the interested reader to consult the quotations in these references. Most of the current work is a brief review of results in Refs. 1-4. The symmetric two-dimensional Ising model is defined by H = - J ^2 (am ,n<^m,n+l + m,,n
<7
m,n°'m+l,n)-
(1)
For this model it is convenient to define elliptic modulus 5 k = l/smh2(2J/kBT),
(2)
which is < 1 for T < Tc and > 1 for T > Tc, with k ->• \/k giving the KramersWannier duality transformation. The spontaneous magnetization is simply given by 6 ' 7 {a)
-{
0,
T>TC.
(3)
The calculation of the pair correlation function C(ra, n) = {(Tofi(Jm,n)
(4)
*This work has been supported in part by NSF Grants No. PHY 97-22159, P H Y 97-24788 and PHY 01-00041. t Email address [email protected] 237
2090
H. Au-Yang
& J. H. H. Perk
is more involved and can be carried out using quadratic difference equations8 [C(m, n+ l)C(m, n - 1 ) - C(m, n)2} +k[C*(m+l,n)C*(m-l,n) - C*(m,n)2]
=0,
(5)
= 0,
(6)
2
[C(m + 1, n ) C ( m - 1 , n) - C(m, n) } +k[C*{m,n+l)C*(m,n-l) - C*(m,n)2]
where C*(m, n) is the dual correlation function obtained by replacing k —> 1/k. For the symmetric case (1), these two equations are equivalent. To solve them we need initial conditions. For T = T c , we have
^"'-"•'"-'-fiixT+fc)-
(7)
which form was already known to Onsager and Kaufman. 9 For T ^ Tc, C(n, n) and C*(n,n) can be calculated by Toeplitz determinants 1 ' 4 ' 7 C(n,n)
= (-1)"
det
({o^-i}),
(8)
l
C*(n,n) =
det
({a^}),
(9)
+ a_ r a )/(2n + 1),
(10)
a - n - i = (2nfca_„+ a„_i)/(2n + 1 ) ,
(11)
where an= (2nk~1an-1
for n = 1,2, • • •, with the initial conditions ao = J ^ [ E ( * ) - ( l - f c 2 ) K ( f c ) ] ,
o_i = ~ E ( f c ) .
(12)
However, it can be done faster by another set of quadratic difference equations due to Jimbo and Miwa. 10 2. High- and Low-Temperature Series for Susceptibility Very recently, with the help of (5) the high- and low-temperature series for the susceptibility were much extended by the authors of Ref. 2. In terms of the reduced susceptibility, oo
X = kBTx=
Yl
(Ko 2 ),
(13)
they found for T > Tc, X= 1 + 4sh + 12sl + 32sl + 764 + ™4 + 4 0 0 s h + • • • + 20073302588291729914311665722841070356623232518453\ 67545550226445723763406738301159160108585998318576 sf3, (14) 238
New Results for Susceptibilities in Planar Ising Models 2091
with s h = s/2 = sinh(2iC)/2, and for T < Tc, X= 4s 4 + 16sf + 104sf + 416s/ 0 + 2224s,12 + • • • + 3051547724509044350855662072500389468463893273907\ 5732810211229434299420849612234517174982030845245\ 5331887458424846630637797467206682914215700492366\ 9271259707379855275224873707435550114462001144064 sf46,
(15)
with Sj = 2/s = 2/sinh(2AT). The size of the coefficients may look ridiculous at first sight. However, it is well-known to series expanders that the new information in each successive coefficient is often in the last few digits. Near the ferromagnetic critical point, the susceptibility behaves asymptotically as P
~'X±
1/2
« C 0 ± (2X c v / 2) 7 / 4 |r|- 7 / 4 F ± + Bu
(16)
where {y/1 + r 2 + r ) 1 / / 2 = 1/^/s and r = (1/s - s)/2, and ± stands for T above or below T c . In (16) the ferromagnetic background is given by 2 B{ = (-0.104133245093831026452160126860473433716236727314 -0.07436886975320708001995859169799500328047632028r -0.0081447139091195995371542858655723893266057740-r2 +0.004504107712232015926355020852986970591364528r3 + 0.16279253648974618861881216566686-r14) +(log|r|)x (0.032352268477309406090656526721221666637730948898r -0.0057755293796884630091487564013201013677152980-r3 + 0.041428586463052869356803144137620T14) 2 +(log|r|) x (0.0093915698711458721317953318727075770649513654r4 -0.00869592546287923802156416645191752987912922T6 H 0.0055571002151161308034896964314679r14) 3 +(log|r|) x (-0.000015771569138451840480001012621461738178T9 +0.0000344282066208887553647799856857753380T11 -0.0000524427177487226174161583779149393r13),
(17)
whereas, the ferromagnetic scaling amplitudes functions are given by2 F+ = 1 + r 2 / 2 - r 4 / 1 2 - 0.1235292285752086663-r6 +0.136610949809095r 8 - 0.13043897213T 1 0 + • • •, for T > Tc, 2
4
6
(18)
F_ = 1 + r / 2 - r / 1 2 - 6.321306840495936623067r +6.25199747046024329r8 - 5.6896599756180r10 + • • •, for T < Tc. (19) 239
2092
H. Au-Yang
& J. H. H. Perk
More coefficients are given in Ref. 2. The last digit in each term above may not be reliable. As we have normalized" F±(T) ->• 1 for T —>• Tc (or r ->• 0), we need to give also the leading susceptibility amplitudes: C(J~ = 1.000815260440212647119476363047210\ 236937534925597789 (2iiT c v^) _ 7 / 4 \/2, CQ = 1.000960328725262189480934955172097\ 320572505951770117 (2 J FsT c v / 2)" 7/4 v^/(127r).
(20)
Near the antiferromagnetic critical point, the susceptibility behaves as r
{VTT^
1%
+ T)
1/2
~ -Baf,
(21)
where Baf =
0.1588665229609474882333592313690210116925239008416 +0.149566836938535905194382029433591286374711207262r +0.01071222587983288033470968550659996768542030678r2 + • • • + 0.007123677682511208149032476379667-r14 +(log|r|) (-0.1553171901580110585934133538932734529992121600305r +0.03206714814586975221843437287457551882247161782r3 + 0.0094056230380765607719474925088649r14) 2 +(log|r|) (0.01153371437882328027949011442761203640684043805r4 -0.011311734920691560067535056532207842716405684T6 + 0.00674470189451526288478200059343432r14) +(log|r|)3 (0.0000578997194764877297760067221144062249541r9 -0.00016991508824012890240796446744935908812T11 +0.00032664884687465587957270016883093909-T13).
(22)
The difference of F+ and F- in Eqs. (18) and (19) implies that a suggestion of Aharony and Fisher 11 breaks down in higher order. They had brought up the possibility that there are "no irrelevant variables." This they concluded from the speculation that the Ising model free energy in the critical region can be described entirely by two nonlinear but "analytic" (thermal and magnetic) scaling fields. Then the scaling amplitude can be found to be
equal above and below T c . 2 a
Note that we have a slight change of notation with respect to Ref. 2, as we have rescaled all B's and F's with a factor t/s. 240
New Results for Susceptibilities
in Planar Ising Models
2093
We now know that this simple picture is incomplete and that corrections to scaling due to breaking of rotational symmetry must be considered. Indeed, the correlation functions have a kind of multipole long-distance expansion,12 which can explain the deviations from fourth order on. Very recently, a conformal field theory explanation has also been given. 13 To study the effect in more detail we shall have to study the model on other lattices. Another interesting feature discussed in Ref. 2 is that the susceptibility has a natural boundary at the critical point, i.e. there exists a closed curve of (essential) singularities fully prohibiting analytic continuation in the complex temperature plane from high to low temperatures. The Ising susceptibility is not differentiably finite, unlike the zero-field free energy and the spontaneous magnetization. This then explains why there is no simple closed form expression available after half a century of research. Yet, we now have algorithms of polynomial complexity, which is as good for numerical analysis. 3. Baxter's Z-invariant inhomogeneous Ising model Baxter's Z-invariant Ising model is defined in terms of a set of oriented straight lines carrying "rapidity" variables Uj, Vj, •••. In the scaling limit the scaled correlation function depends on a single distance variable R, as first discovered by Bai-Qi Jin, 1 2m
*-5
^ 2
I ^cos(2Uj)l
2m
(
>, 2-i 1/2
+ij>n(2^)l
.
(24)
' 3=
This is given in terms of the 2m rapidity variables crossing between the two spins in question. Using the diagonal correlation length £d to introduce the scaled distance r = R/U,
where
^
= \\ogk\,
(25)
we have found the most general form of the scaled correlation functions to be K ) « |1 - k-2\l'AF{r),
{aa')* » |1 - Jfe- 2 | 1/4 G(r),
(26)
where the functions F(r) and G(r) satisfy FF" - F'2 = - r _ 1 G G ' ,
GG" - G'2 = -r^FF',
(27)
and the front factor is the square of the spontaneous magnetization for T < Tc or k > 1. F(r) and G(r) are the Painleve functions for the uniform rectangular Ising lattice, 14 see Refs. 1,4 for more details. 4. Susceptibility in Z-Invariant Lattice For a general ferromagnetic Z-invariant lattice with J\f sites, the susceptibility \ is given by X = & B Tx = ^lirn^-^ ^
«^ml,mO,ma,ni,) - N , o ) 2 ) ,
^
m\,ni n%2,n2 241
(28)
2094
H. Au-Yang
& J. H. H. Perk
where ( m i , n i ) and {1712,712) run through the possible coordinates of the spins. In periodic cases one of the two sums can be done trivially. In quasiperiodic cases this can only be done asymptotically at the largest distance scale. Hence, in the scaling limit and for both periodic and quasiperiodic ^-invariant lattices, x becomes
j—00
J—00
"•
where ^
l
=
\l-k-^(G(R/^)-l),
^ ^ = |l-fc-2rFWed),
(30)
(31)
K = l/£d = I log A;I, and R reduces to R = VaM2 + 2bMN + cN2
(32)
with a, b, and c known constants that can be calculated choosing suitable integer coordinates M and N. Also, go is the corresponding multiplicity factor counting how many spin distance vectors fall exactly or asymptotically within a unit cell in the (M, N) plane. Therefore,4 X = ~^t== / ° ° d r r 3 / 4 F ± ( r ) *~ 7 / 4 + • • • = A±\t\~^ + 0(\t\^% (33) Vac — \r Jo with t = \T — Tc\/Tc, giving the exact T > Tc and T
(34)
Also, more generally, X = - 7 ^ = I log(fc)|- 7 / 4 r ° d r r » / ^ ± ( r ) (35) Vac — oJ Jo is a product of a factor depending on rapidities and the modulus and a factor which is a universal integral over a Painleve V function. Hence, the amplitudes 242
New Results for Susceptibilities in Planar Ising Models 2095 are known—in principle—to this high accuracy for all Z-invariant (quasi)-periodic cases. We plan to use these values later to analyze long series for t h e isotropic triangular and honeycomb lattices, once they are available. We note t h a t the numbers given above agree to a few places w i t h earlier series extrapolations. Four of the six agree to about ten places w i t h those of W u et al. 1 4 and of Vaidya. 1 5 For T above T c , they agree to b e t t e r t h a n t h r e e places with those obtained from the Syozi-Naya 1 6 approximation, b u t this c a n b e u n d e r s t o o d as this approximation is precisely the x< approximation in W u et al. 5. O u t l o o k We are working to extend and analyze series for other lattices in order to get more information on irrelevant variables in t h e corrections t o scaling, having a preliminary algorithm of polynomial complexity for t h e isotropic honeycomb a n d triangular lattices which reproduces the known series coefficients. B u t more work needs to be done to increase its efficiency, as we will need t o go t o one to two hundred terms, before being able to see clearly t h e effect of t h e irrelevant variables. We are also looking at the susceptibility of Ising models on Penrose tilings. Finally, we also want to look at the effect of frustration, which occurs in t h e regime where elliptic modulus k is purely imaginary. References 1. H. Au-Yang, B.-Q. Jin, and J.H.H. Perk, J. Stat. Phys. 102 (2001) 501-543. 2. W.P. Orrick, B.G. Nickel, A.J. Guttmann, and J.H.H. Perk, J. Stat. Phys. 102 (2001) 795-841; Phys. Rev. Lett. 86 (2001) 4120-4123. The complete set of series coefficients is posted at http://www.ms.unimelb.edu.au/~tonyg . 3. H. Au-Yang and J.H.H. Perk, in MathPhys Odyssee 2001: Integrable Models and Beyond, T. Miwa and M. Kashiwara, eds., (Birkhauser, Boston, 2002), pp. 1-21. 4. H. Au-Yang and J.H.H. Perk, in MathPhys Odyssee 2001: Integrable Models and Beyond, T. Miwa and M. Kashiwara, eds., (Birkhauser, Boston, 2002), pp. 23-48. 5. L. Onsager, Phys. Rev. 65, 117-149 (1944). 6. L. Onsager, Nuovo Cimento (ser. 9) 6 (Sup.), 261 (1949); C.N. Yang, Phys. Rev. 85, 808-816 (1952). 7. B.M. McCoy and T.T. Wu, The Two-Dimensional Ising Model, (Harvard Univ. Press, Cambridge, Mass., 1973). 8. J.H.H. Perk, Phys. Lett. 79A (1980) 3-5. 9. B. Kaufman and L. Onsager, Phys. Rev. 76, 1244-1252 (1949). 10. M. Jimbo and T. Miwa, Proc. Japan Acad. 56A, 405-410 (1980); 57A, 347 (1981). 11. A. Aharony and M.E. Fisher, Phys. Rev. Lett. 45, 679-682 (1980); Phys. Rev. B 27, 4394-4400 (1983). 12. H. Au-Yang and J.H.H. Perk, Phys. Lett. A 104, 131-134 (1984); X.-P. Kong, Ph.D. thesis, Stony Brook (1987). 13. M. Caselle, M. Hasenbusch, A. Pelissetto, and E. Vicari, cond-mat/0106372. 14. T.T. Wu, B.M. McCoy, C.A. Tracy, and E. Barouch, Phys. Rev. B 13, 316-374 (1976). 15. H.G. Vaidya, Phys. Lett. A 57, 1-4 (1976). 16. I. Syozi and S. Naya, Progr. Theor. Phys. 24, 374-376 (1960).
243
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2097-2106 © World Scientific Publishing Company
ALGEBRAIC G E O M E T R Y A N D H O F S T A D T E R T Y P E MODEL
SHAO-SHIUNG LIN Department
of Mathematics,
Taiwan
University,
Taipei, Taiwan *
SHI-SHYR ROAN t Institute of Mathematics,
Academia
Sinica, Taipei, Taiwan *
Received 15 October 2001 In this report, we study the algebraic geometry aspect of Hofstadter type models through the algebraic Bethe equation. In the diagonalization problem of certain Hofstadter type Hamiltonians, the Bethe equation is constructed by using the Baxter vectors on a high genus spectral curve. When the spectral variables lie on rational curves, we obtain the complete and explicit solutions of t h e polynomial Bethe equation; t h e relation with the Bethe ansatz of polynomial roots is discussed. Certain algebraic geometry properties of Bethe equation on the high genus algebraic curves are discussed in cooperation with the consideration of the physical model.
1. Introduction It is known for the past decade that algebraic geometry has played a certain intriguing role in certain 2-dimensional solvable statistical lattice models, a notable example would be the chiral Potts iV-state integrable model (see e.g., Refs. 1,3 and references therein). In the note, we report the algebraic geometry aspect of another model of physical interest in solid state physics. In the early 90's, motivated by the work of Wiegmann and Zabrodin 12 on the appearance of Uq{sl2) symmetry in problems of magnetic translation, Faddeev and Kashaev 6 pursued the diagonalization problem on the following Hamiltonian by the quantum transfer matrix method which was developed by the Leningrad school in the early eighties: HFK = n{aU + a^U'1)
+ u(pV + p^V'1)
+ P(jW
+ 7 " 1 W~l)
,
(1)
where U, V, W are unitary operators with the Weyl commutation relation for a primitive iV-th root of unity OJ and the iV-th power identity property, UV = uiVU, VW = wWV, WU = wUW; UN = VN = WN = 1. As a special limit case for p = 0, the model is reduced to the (rational flux) Hofstadter Hamiltonian, a model *e-mail: [email protected] t Supported in part by the NSC grant of Taiwan. t e-mail: [email protected]. edu.tw 245
2098
S.-S. Lin & S.-S. Roan
possessing several physical interpretations with the history which can trace back to the work of Peierls 11 on Bloch electrons in metals with the presence of a constant external magnetic field. By the pioneering works of the 50s and 60s, 2 ' 5 , 7 ' 9 ' 1 3 the role of magnetic translations was found, and it began a systematic study of this 2D lattice model. In 1976, Hofstadter 8 found the butterfly figure of the spectral band versus the magnetic flux which exhibits a beautiful fractal picture. Here the phase of u> represents the magnetic flux (per plaquette). In Ref. 6, a general frame work to determine the eigenvalues of certain quantum chains appeared in the transfer matrix was presented. The method relies on a special monodromy solution of the Yang-Baxter equation for the six-vertex i?-matrix; this solution appeared also in the study of chiral Potts model. 3 For a finite size L, the trace of the monodromy matrix gives rise to the transfer matrix acting on the quantum space ® C ; while the Hofstadter type Hamiltonian (1) can be realized in the case L = 3. In general, the diagonalization problem of the transfer matrix can be formulated into the Bethe equation through the Baxter vector3, visualized on a "spectral" curve associated to the corresponding model. In Ref.,10 we presented a detailed and rigorous mathematical study on the Bethe equation associated to the Hofstadter type model. In particular, we obtained the complete solution of the Bethe equation for models with rational spectral curves for L < 3, among which a special Hofstadter type of HFK in Ref. 6 is included, and further expended to all the other sectors. In this note, we explain the main results we have obtained in Ref. 10; detailed derivations, as well as extended references to the literature, may be found in that work. This paper is organized as follows. In Sect. 2, we first recall results in transfer matrix relevant to our discussion; then introduce the Bethe equation (or Baxter T-Q equation) through the Baxter vector on the spectral curve. In Sect. 3, we consider the case when the spectral data lie on rational curves and perform the mathematical derivation of the answer. We present the complete solutions of the Bethe polynomial equations of all sectors for L < 3. In Sect. 4, we discuss the "degeneracy" relation between the Bethe solutions and the eigenspaces in the quantum space of the transfer matrix for L = 3; also its connection with the usual Bethe ansatz technique in literature, in particular the result obtained in Ref. 6. In Sect. 5, we describe the algebraic geometry properties of the high genus spectral curve arisen from the Hofstadter Hamiltonian. N o t a t i o n s . The letters Z, R, C will denote the ring of integers, real, complex numbers respectively, N — Z>o, ZJV = Z/iVZ. Throughout this report, N will always denote an odd positive integer with M = [y]: N = 2M + 1, M > 1; u) is a primitive ./V-th root of unity, and q := w5 with qN = 1, i.e., q = w M + 1 . An element v in the vector space CN is represented by a sequence of coordinates, Vk, k G Z, with the iV-periodic condition, Vk = Vk+N, i.e., v = (vk)ke2, • The standard basis of C ^ will be denoted by \k), with the dual basis of C^* by (k\ for k G ZJV. For a positive
a
It is also called as the "Baxter vacuum state" in other literature. 246
Algebraic Geometry and Hofstadter
Type Model
2099
integers n, we denote ® C the tensor product of n-copies of the vector space CN. We use the notation of p-shifted factorials: (a; p)n = (1 — a)(l — ap) •••(!.— a p n _ 1 ) for n € N, and (a; p)o = 1. 2. Transfer Matrix and t h e B e t h e Equation We consider the Weyl algebra generated by the operators Z, X satisfying the Weyl commutation relation with the 7V-th power identity, ZX = CJXZ, ZN = XN = I, and denote Y := ZX. In the canonical irreducible representation of the Weyl algebra, the operators Z,X,Y act on C with the expressions: Z(v)k = wkVk, X{v)k = Vfc-i, Y{v)k = oJkVk-\- It is known that the following L-operator for an element h = [a : b : c : d] of the projective 3-space P with operator-entries acting on the quantum space C ^ , r
, ,
L x)=
( aY xbX\
^ {xcZ
_
XeC
d J'
>
possesses the intertwining property of the Yang-Baxter relation, R(x/x')(Lh(x)(g)l)(l
= (l(g)Lh(x'))(Lh(x)(g)l)R(x/x')
aux
aux
, (2)
aux
where R(x) is the matrix of a 2-tensor of the auxiliary space C 2 with the following numerical expression,
J?(x)
(xw-x'1 0 0 \ 0
0 u}{x-x~l) w-1 0
0 w-1 x-x'1 0
0 \ 0 0 xuj-x'1/
By performing the matrix product on auxiliary spaces and the tensor product of quantum spaces, one has the L-operator associated to an element h = (ho, • • •, / » L - I ) e ( P 3 ) L , L^(x) = 0 j j o Lhj(x), which again satisfies the relation (2). The entries L
N
of L^(x) are operators of the quantum space (g) C , and its trace defines the commuting transfer matrices for x G C, T^(x) = tiaux(L^(x)). The transfer matrix T%(x) can also be computed by changing L^ to L\lj via a gauge transformation: ^(Z.&.CJ+I) AL
= AjLhj(x)Ajl1,0
< j < L- 1, with A, = L
7
)
and
:= AQ. One has
where Fh(x,^^')
:= £aY - xbX + g&cZ
- id. Hence TK(x) =
£••= ( £ O , - - - , £ L - I ) where
J=0
^ 2 , 1 ( ^ 0 Ln#,2(x> 0. 247
trami(LR(x,$),
2100 S.-S. Lin & S.-S. Roan
We consider the variables (x, £o, • • • > £ L - I ) in the following spectral curve, fN
C--
N _
NhN
fi*r-(-1)1*22+121 ^• +1 a; cj
°i aj
j-Q
r-l
and denote pj = (x,£j,£j+i). Then the operator Fhj(x,£j,£j) null space in C generated by the vector \pj) with the form: ( m IPj)
/Olo \ = 1 U W
'
=
(m-l\pj)
(3)
has 1-dimensional
^+iQJ^W ~ ^ j
-Zjiti+itW"
- dj) •
The Baxter vector \p) for p £ C^ is defined by |p) := \po) . . . ® | P L - I ) 6® C N , which possesses the following property: ^ L i f o ^ l P ) = k-P>A_(p), ^ . 2 , 2 0 z , | ) | p > = \r+p)A+(p)
, ZK.2jl(a:,£)|p) = 0 ,
where A± are functions of C^ defined by A_ (x, £) = Ylj=o (dj—x£j+icj), J
Cj
an(
T
are tne
A+(a;,£) =
,±1
rijM) g° /a~-zb- ' ^ ± automorphisms, r±(a;, £) = (q s, 9 - 1 ^)- It follows the important relation of the transfer matrix on the Baxter vector over the curve Cfr, TK(x)\p) = |r_p)A_(p) + \T+p}A+(p) , for p e CK .
(4)
As Tjfcc) are commuting operators for x € C, a common eigenvector {<^| is a constant vector of C with an eigenvalue A(ar) € C[x\. Define the function Qip) = (
(5)
By the definition of T^(x),A(x), one can easily see that T%(x) is an operatorcoefficient even x-polynomial of degree 2[Lr] with the constant term To = fT Jo aj 0 7 + EL Jo dj. Hence the polynomial A(x) in (5) is an even function of degree < 2[j] with A(0) = ql Ylj=0 aj + Ylj=o dj for some I € ZN. For L = 3, we have Tn(x) =T0 + x2T2 where T2 = b0cia2X ® Z (g) y + a 0 bic 2 y 0 X Z + c0aib2Z ® y X + co&icfo-Z ® X ® / + doc^I ®Z ®X + b0d1c2X ®I®Z .
. . '
The above T2 can be put into the form of the Hofstadter type Hamiltonian (l). 6 , 1 0 In the equation (5), Q(p) is a rational function of Cr with zeros and poles. Hence the understanding of the Bethe solutions of (5) relies heavily on the function theory of CJI, and the algebraic geometry of the curve inevitably plays a key role on the complexity of the problem. 248
Algebraic Geometry and Hofstadter
Type Model
2101
3. The Rational Degenerated Bethe Equation In this section, we consider the case when the spectral curve C% degenerates into an union of rational curves under the conditions:^ = q~xdj, bj = q~xCj for j = 0, ...,L — 1. By replacing Cj,dj by ^ - , 1 , we assume dj = 1 for all j with the parameter CjS to be generic. In this case, C^ is the union of disjoint copies of the a;-(complex) line, containing the following r±-invariant subset of C^ which will be sufficient for the discussion of Bethe equation, C := {{x, & , . . . ,£ L -i)l£o = • • • = & - 1 = ql, I € ZN}. We shall make the identification C = P 1 x ZJV via (x,ql,... ,ql) -f-> (a;,I). The automorphisms T± on C become T±(X,I) = (q±1x,I — 1), by which the action (4) of T(x)(:= T^(x)) on the Baxter vector \x, I) now takes the form, T(x)\x,l)
= \q-1x,l-l)A-(x,l)
+ \qx,l-l)A+(x,l)
,
(7) 1
where A± are the rational functions of x:A-(x,l)
= Ylj=0 (1 — xcjq ), A+(x,l)
=
T\j=a i-xc q~l • Furthermore, one can express the Baxter vector \x, I) over the curve —A —2.
I
— l-\
C in the component-form: (k\x, I) = q^ JJjZo Lc- i+*-u)—~i- Here the bold letter k denotes a multi-index vector k = (A;o,... , & L - I ) for kj £ ZN with the squarelength of k defined by |k| 2 := ^ , C 0 k2. Each ratio-term in the above right hand side is given by a non-negative representative for each element in ZJV appeared in the formula. We have the following result on the Bethe equation and its connection with the transfer matrix T(x): Theorem 1: Denote fe, f° the functions on C, fe(x, 2n) = U^} •F o r
f°(x, 2n + 1) = Uf=o ^{xc-ZZlT L vectors in £g> C
x e p l
'l
e ZN
'
(
^! J ; a '"x ) " +1 , and
w e define the
folkwing
N
,
JV-1
\x)f = J2 \x> 2 n ) / e ( x , 1n)Jn
,
la;)? = E ^ " 1 \x, 2n + l ) / ° ( x , 2n + l)Jn
,
n=0
\x)t
= \x)fq~lu(qx)
+ \x)°u(x);
where u{x) := Uj^oi1
~ xN cf){xcjq;q2)M
Then (i) \x)fu(qx) = \x)°qlu(x), or equivalently, |ai);+ = 2q~l\x)fu(qx) (ii) The T(a;)-transform on |a;)z+ is given by q~lT(x)\x)+
= |g- 1 x)+A_(a;,-l) + |ga ; )+A + (a ; ,0) ,
.
= 2|a;}°w(a;).
I £ ZN .
(iii) For a common eigenvector (\x)f) and A(a?) are polynomials with the properties: deg.Q^~(a:) < (3M + l)L , deg.A(a;) < 2[f ], A(x) = A(-a;), A(0) = q21 + 1, and the following Bethe equation holds: i-l l
q- A(x)Q+(x)
L-l 1
= ] } ( 1 - MiQ-^QtixQ- ) 249
+ U.0- + xCj)Qt(xq)
.
(8)
2102
S.-S. Lin & S.-S.
Roan
Furthermore for 0
<
m
<
Qm(x),Qti-m(x)
M,
are
elements
in
By (iii) of the above theorem, the equation (8) for the sector m,N — m can be combined into a single one. For the rest of this report the letter m will always denote an integer between 0 and M: 0 < m < M. By introducing the polynomials Am{x),Q{x) via the relation, L-l
{Am{x), xm H(l-xNc?)Q(x))
= (q-mA(x),
(qmA(x),
Q+(x)),
Q+N_m{x))
,
3=0
the equation (8) for I = m,N — m becomes the following polynomial equation of Q(x),Am(x): L-l m
Am{x)Q{x) = q~
L-l 1
m
J J (1 - xcjq^Qixq- )
+q
J ] (1 + xCj)Q{xq)
,
(9)
with deg.Q(a;) < ML-m, deg.Am(ai) < 2[f-], Am(a;) = A m ( - a ; ) , A m (0) = qm + m q~ . The general mathematical problem will be to determine the solution space of the Bethe equation (9) for a given positive integer L. For L = 1,2, we have the following result. Theorem 2: (I) For L = 1, we have Am(x) = qm + q~m and the solutions Qm(x) of (9) form an one-dimensional vector space generated by the following polynomial of degree M — m, M m
~
Bm(x)
i
nm+i-l
_
g
„-m-i
= i + £ (IIq„m mi+ q-._m JL-i j=i
^+JW
i=i
(II) For L = 2, the equation (9) has a non-trivial solution Qm(x) if and only if deg.Qm(a;) = M — m + m' for 0 < m' < M. For each such m', the eigenvalue + g- m '- 2 )a; 2 CoCi + qm + q~ and Am(x) in (9) is equal to Am<m>(x) qi(q™'~i the corresponding solutions of Qm(x) form an one-dimensional space generated by a polynomial Bmtmt(x) of degree M — m + m' with -Bm>m-(0) = 1. For L = 3, this is the case related to the Hamiltonian (1). We consider the N x N matrix, / <5w-i U'N-I jV-2
w N-3
0
•••
u
S'iV-2 f
"iV-2
0
u V N-3
uS'N
N-3
0 0 \
U "JV-3
(10)
»i ^i u[ 0
\ o 250
w'0 v'0 6'0 j
Algebraic Geometry and Hofstadter Type Model 2103
with the entries defined by w'k — qk+ 2 +q k 2—qm—q m,v'k = (qk+*—q k ci+c 2 ), S'k = (qk~i +q~k~i){c0c1+c1c2 + c2Co), u'k = (qk-§ -q-k~§)c0c1c2. one can derive the following result.
2
)(co+ Then
Theorem 3: For L = 3, the condition of the eigenvalue Am(a;) = \mx2+qm+q~m, 0 < m < M, with a non-trivial solution Qm(x) in the equation (9) is determined by the solution of det(A — A m ) = 0, where A is the matrix defined by (10). For each such Am(x), there exists an unique (up to constants) non-trivial polynomial solution Qm{x) of (9) with the degree Qm(x) equal to 3M — m and Q m (0) ^ 0. 4. The Degeneracy and B e t h e Ansatz Relation of R o o t s of Bethe Polynomial We first discuss the degeneracy relation of eigenspaces of the transform matrix T(x) L
N
in <8> C * with respect to the Bethe solutions obtained in the previous section. As before, we denote A(x) the eigenvalues of T{x), whose constant term is given by L
T0 = D + l, where D := q~L Y; hence A(0) = qL + 1. For I G ZN, we denote E;L the iV L_1 -dimensional eigensubspace of ® CN* of the operator D with the eigenvalue ql. For 0 < m < M, the equation (9) describes the relation of A(x) and its eigenfunctions with A(0) = q2m + 1 or ^ 2 ( J V _ m ) + 1. We now consider the case for L — 3, where T2 in (6) is now expressed by T2 = q~2{cQcxX Z ® Y + cxc2Y ®X®Z + CQC2Z +q-1(c0c1Z ® X / + cic 2 7 ® Z ® X + CQC2X
®Y®X) ®I®Z).
We have qD = {Z®X®I){X®I®Z){I®Z®X). We shall denote O3 the operator algebra generated by the tensors of X, Y, Z, I appeared in the above expression of T2. Then O3 commutes with D, hence one obtains a 03-representation on E 3 for each /. With the identification, U = D~^2Z® X® I, V = D'^^X®I®Z,0^is generated by U, V which satisfy the Weyl relation UV = uVU and the JV-th power identity. Hence O3 is the Heisenberg algebra and contains D as a central element. Then qD~T2 has the following expression, coCl{U + U'1) + c0c2(V + V-1) + Clc2(qD5/2UV
+ q^D-^V^U'1)
. 1
(11)
The above Hamiltonian is the same as HFK (1) with W — q~ D~ / V~ U~1, a = /3 = 7 = 1. Our conclusion on the sector m = M is equivalent to that in Ref. 6 as it becomes clearer later on. There is an unique (up to equivalence) non-trivial irreducible representation of O3, denoted by Cp , which is of dimension N. For each I, E3 is equivalent to AT-copies of C ^ as £>3-modules: E 3 ~ N C%. For 0 < m < M, we consider the space E 3 with ql = q±2m. The evaluation of E 3 on |x)±TO gives rise to a AT-dimensional kernel in E 3 . By Theorem 3, there are N polynomials Qm(x) of degree 3M - m as solutions of (9) with the corresponding N distinct eigenvalues Am(ar). The AT-dimensional vector space spanned by those Qm(x)s becomes a realization of the irreducible representation Cp for the Heisenberg algebra O3. 251
5 2
1
2104
S.-S. Lin & S.-S.
Roan
Now we discuss the relation between the Bethe equation (9) and the usual Bethe ansatz formulation in literature. For 0 < m < M, a solution Qm(x) in (9) always have the property Q m (0) ^ 0 by Theorem 3, hence one has the form a 3M-mQm(x) = U^=i~m(x ~ Tt) w i t h zi £ c * % setting x = zf1 in (9), we obtain the following relation among 2/s, which is called the Bethe ansatz relation, 2 < r +
,
3M-m
!TTiL±£L=
f^zi-Cj
TT
SfLli",
J^zi-qzn
t
< /<
3 M
-m .
For the sector m = M, the comparison of the ^-coefficients of (9) yields the expression of eigenvalue, 2M
AM = (q^~ +q~)s2
+ (qi - q~)si ^ z
n
+ (qi + q~ - q? -q~)^zizn
n=l
. Kn
With the substitution, p, = q^c^1^ = q^c^[l,p = q^c^1, the above expression coincides with (5.27) in Ref. 6. Note that the Bethe ansatz relation can be shown to be equivalent to the Bethe equation (9) for the sector M. However, the parallel statement is no longer true for other sectors m ^ M, i.e., it does exist some nonphysical Bethe ansatz solutions in the above form, while not corresponding to any polynomial solution of Bethe equation (9). Some example can be found in the (M - l)-sector. 5. High Genus Curves for the Hofstadter Model We are now going back to the general situation in Sect. 2. Note that the values ^ s of the curve C^ in (3) are determined by £Q and xN, denoted by y = xN, r] = $ . The variables (y, rj) defines the curve which is a double cover of y-line, % = Cn(yW where the functions A^,B^,CR,DR
+ (AR(y) - DR(y))V
- BR(y) = 0
are the following matrix elements,
(-At{v) B%(y) \ Jyf (-a? yb? \ Now we consider only the case: L = 3, ao = do = 0, bo = Co = 1, with generic hi, h2. The expression of T(x) is given by T(x) = x2(cia2X
®Z®Y
+ axb2Z ® Y ® X + bxd2Z ® X i" + dxc2X I ® Z),
equivalently, x~2D~T(x) is equal to the Hofstadter Hamiltonian (l) p =o with U = D~l/2Z X I, V = D'1/2X ® I ® Z and //, v, a, (3 related to hi, h2 by \i2 = qb\Cia2d2, a2 = q~xb\c^ 1a2~1d2, v2 = qa,\dib2C2, /32 — q~1a,i1dib2~1C2. By factoring out the y-component of B^, the main irreducible component of B^ is the curve, B : (y 2 6f c»+a»a»)Tj2 + (a»b»+b?d»-c»a»-d?c»)yr,-(y2c?&^+«) 252
= 0,
Algebraic Geometry and Hofstadter Type Model 2105
which is an elliptic curve as a double-cover of the y-line. For the curve C^, the variables £o and £1 are related by ^ = £J~N, which implies that Cg can be identified with W x ZAT where W is a genus 6N3 — 6iV2 + 1 curve with the following equation in the variable p — (x, £o, £2)5 -tNaN .
w KV -
t-N ? 0
_
4- rNhN
S2 "l + x °1 _ JN ' TNj:NrN c X c; 2 l "l
-PNnN ^JV _ S2
4-
SO " 2 ^ „N cN„N x SO c 2
x
°2 JN ' 2
— a
By averaging the Baxter vectors \p,s) of C^ over an element p of VV, \p) := 1? S s =o IP> s)ls ' which defines the Baxter vector on W. Furthermore, the transfer matrix can be descended to one on W with the following relation, x-2T(x)\p) = |T_(p))A_(p) + |r + (p))A+(p) , where A± are the functions on W: A_(z, £0,6) = ( 8 6 C l - * j g o C a - < f a > , A+(x, &, 6 ) = g 2 ( ^ \ t a : - X t t - x t ) 2 C 2 ) - F o r a n eigenvector (
(£>£<)) £2)- Consider the -D-eigenspace decomposition of CN* = © i G z ^ 3 - ^he evaluation of the Baxter vector over W gives rise to the following linear transformation, Ei : E 3 —> {rational functions of W} with si(v)(p) : = (v\p), for I € ZN. One has the following result. Theorem 4: For I € ZJV, the linear map ei is injective, hence it induces an identification of E 3 with a 7V2-dimensional functional space of W. By the discussion in Sect. 4, as the Heisenberg algebra O3 representations , E 3 is equivalent to TV copies of the standard one. Hence it induces an 03-module structure on the function space £j(E 3 ), induced by the one of E 3 by above theorem. The mathematical structure of the functional space £;(E 3 ) in terms of the divisor theory of Riemann surfaces in corporation with the interpretation of Heisenberg algebra representation remains an algebraic geometry problem for further study. Acknowledgements S.-S. Roan is grateful to M. Jimbo, J. Kellendonk, B. M. McCoy, R. Seiler for fruitful discussions. This work was reported in the workshop "Quantum Algebra and Integrability" CRM Montreal, Canada, April 2000, and was a subject of an Invited Lecture at APCTP-Nankai Joint Symposium on "Lattice Statistics and Mathematical Physics", Tianjin, China, October 2001, to which he would like to thank for their invitation and hospitality. References 1. G. Albertini, B. M. McCoy, and J. H. H. Perk, in Adv. Stud. Pure Math., vol. 19, (Kinokuniya Academic 1989) p.l. 253
2106 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
S.-S. Lin & S.-S. Roan M. Ya. Azbel, Sov. Phys. JETP 19, 634 (1964). R. J. Baxter, V. V. Bazhanov and J. H. H. Perk, Int. J. Mod. Phys. B4, 803 (1990). V. V. Bazhanov and Yu. G. Stroganov, J. Stat. Phys. 59, 799 (1990). W. G. Chambers, Phys. Rev. A140, 135 (1965). L. D. Faddeev and R. M. Kashaev, Comm. Math. Phys. 155, 181 (1995), hepth/9312133. P. G. Harper, Proc. Phys. Soc. London A 6 8 , 874 (1955). D. R. Hofstadter, Phys. Rev. B14, 2239 (1976). D. Langbein, Phys. Rev. 180 , 633 (1969). S. S. Lin and S. S. Roan, "Algebraic geometry approach to the Bethe equation for Hofstadter type models", cond-mat/9912473. R. Peierls, Z. Phys. 80, 763 (1933). P. B. Wiegmann and A. V. Zabrodin, Nud. Phys. B422, 495 (1994); JVucL Phys. B451, 699 (1995), cond-mat/9501129. J. Zak, Phys. Rev. A 1 3 4 , 1602 (1964); Phys. Rev. A134, 1607 (1964).
254
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2107-2112 © World Scientific Publishing Company
LIMITATIONS O N Q U A N T U M C O N T R O L
ALLAN I. SOLOMON* Laboratoire de Physique Theorique des Liquides,
University
of Paris VI, France
SONIA G. SCHIRMERt Quantum Processes Group, Open University Milton Keynes MK7 6AA, United Kingdom
Received 17 October 2001 In this note we give an introduction to the topic of Quantum Control, explaining what its objectives are, and describing some of its limitations.
1. What is Quantum Control? The objectives of Quantum Control are • To determine quantum mechanical systems which will drive an initial given state to a pre-determined final state, the target state. • To describe—and hopefully implement—quantum systems which will through time evolution optimize given operator expectations corresponding to observables of the system. Among a wealth of applications are those to Quantum Computing, where it is clearly essential to be able to start off a quantum procedure with a given initial state, and to problems involving the population levels in atomic systems, such as the laser cooling of atomic or molecular systems. The mathematical tools necessary for the theoretical investigation of these control problems are diverse, involving algebraic, group theoretic and topological methods. The questions that one may ask include: (i) When is a given quantum system completely controllable? (ii) If a system is not completely controllable, how does this affect optimization of a given operator? (iii) How near can you get to a target state for a not completely controllable system? * Permanent address: Quantum Processes Group, Open University, Milton Keynes MK7 6AA, United Kingdom. Email: [email protected] tEmail: [email protected] 255
2108
A. I. Solomon
& S. G.
Schirmer
The answer to the first question depends on a knowledge of the Lie algebra generated by the system's quantum hamiltonian, that to the second arises from properties of the Lie group structure, while the last clearly involves ideas of topology. Especially in the area of Lie group theory, there is a large corpus of classical mathematics which can supply answers to questions arising in quantum control. In particular, for the type of controllability known as Pure State Controllability classical Lie Group theory has already given the basic results. The quantum control system we shall consider is typically of the form M
H = H0 + Y, fm(t)Hm,
(1)
m=l
where Ho is the internal Hamiltonian of the unperturbed system and Hm are interaction terms governing the interaction of the system with an external field. The dynamical evolution of the system is governed by the unitary evolution operator U(t,0), which satisfies the Schrodinger equation ih-U(t,0)=HU(t,0)
(2)
with initial condition {7(0,0) = I, where / is the identity operator. By use of the Magnus expansion, it can be shown that the solution U involves all the commutators of the Hm. The operators Hm, 0 < m < M, in (1) are Hermitian. Their skew-Hermitian counterparts iHm generate a Lie algebra L known as the dynamical Lie algebra of the control system which is always a subalgebra of u(N), or for trace-zero hamiltonians, su(N). The degree of controllability is determined by the dynamical Lie algebra generated by the control system hamiltonian H. If L = u(N) then all the unitary operators are generated and we call such a system Completely Controllable. A large variety of common quantum systems can be shown to be Completely Controllable. 1 ' 2 The interesting cases arise when L is a proper subalgebra of u(N). Such systems may still exhibit Pure State Controllability, in that starting with any initial pure state any target pure state may be obtained, as distinct from the Completely Controllable case, when all (kinematically admissible) states—pure or mixed—may be achieved. 2. Pure s t a t e controllability We shall restrict our attention here to finite-level quantum systems with JV discrete energy levels. The pure quantum states of the system are represented by normalized wavefunctions |\P), which form a Hilbert space H. However, the state of a quantum system need not be represented by a pure state |$) € H. For instance, we may consider a system consisting of a large number of identical, non-interacting particles, which can be in different internal quantum states, i.e., a certain fraction w\ of the particles may be in quantum state | * i ) , another fraction w^ may be in another state |*2) and so forth. Hence, the state of the system as a whole is described by a discrete ensemble of quantum states \^n) with non-negative weights wn that sum 256
Limitations
on Quantum
Control
2109
up to one. Such an ensemble of quantum states is called a mixed-state, and it can be represented by a density operator po on H with the spectral representation N
P0 = Y^Wn\^n)(^n\,
(3)
n=l
where {\$n) '• 1 < n < TV} is an orthonormal set of vectors in H that forms an ensemble of independent pure quantum states. The evolution of po is governed by p(t) = U(t,0)PoU(t,Q)\
(4)
with U(t, 0) as above. Clearly if all the unitary operators can be generated we have the optimal situation, complete controllability. However, classical Lie group theory tells us that even when we only obtain a subalgebra of u(N) we can obtain pure state controllability. The results arise from consideration 3 of the transitive action of Lie groups on the sphere Sk. The classical "orthogonal" groups 9(n, F) where the field F is either the reals SR, the complexes C or the quaternions %, are denned to be those that keep invariant the length of the vector v = {v\,V2, • • • ,vn); the squared length is given by v^v = £]™=1iTO, where Hi refers to the appropriate conjugation. These compact groups are, essentially, the only ones which give transitive actions on the appropriate spheres, as follows: (i) 9(71,3?) = 0(n) transitive on S^""1) (ii) Q(n,C) = U(n) transitive on S(2n~V (iii) ®{n,U) = Sp{n) transitive on Sf4™-1). Since we may regard our pure state as a normalized vector in CN and thus as a point on S^2N~l\ we obtain pure state controllability only for U(N) (or SU(N) if we are not too fussy about phases) and Sp(N/2), the latter for even N only. (Note that we cannot get 0(2N) as a subalgebra of U(N).) Complete controllability is clearly a stronger condition than pure state controllability. To illustrate our theme of the limitations on quantum control, we now give two examples based on a truncated oscillator with nearest-level interactions for which the algebras generated are so(N) and sp(N/2). Both these examples are generic.
3. Examples 3.1. Three-level oscillator
with dipole
interactions.
Consider a three-level system with energy levels E\, Ei, E3 and assume the interaction with an external field /1 is of dipole form with nearest neighbor interactions only. Then we have H = HQ + f(t)H\, where the matrix representations of HQ and 257
2110 A. I. Solomon & S. G. Schirmer
Hi are H0 =
Ei 0 0 " 0 E2 0 , 0 0 E3_
HX =
" 0 di 0 di 0 d2 . 0 d2 0
If the energy levels are equally spaced, i.e., E2 — E\ = E3 — E2 = /x and the transition dipole moments are equal, i.e., d\ = d2 = d then we have H'Q
-10 0 0 00 0 01
V
010 101 010
Hx
where H'Q is the traceless part of HQ. Both \H'0 and \H\ satisfy A + Af = 0
A J + JA = 0
where "0 0 1" 0 -10 .1 0 0.
J
which is a defining relation for so(3). The dynamical Lie algebra in this case is in fact so(3). It is easy to show that the matrix B = UAU^ is a real anti-symmetric representation of so(3) if U — U*J. Explicitly, a suitable unitary matrix is given by
u
1/V20
1/V2'
i/y/2 0
-i/y/2
0
i
0
Since the dynamical algebra and group in the basis determined by U consists of real matrices, real states can only be transformed to real states; this means that for any initial state there is a large class of unreachable states. This example is generic as it applies to iV-level systems, although for even N we need other than dipole interactions to generate so(N). The analogous dipole interaction generates sp(N/2) in the even N case, as we now illustrate.
3.2. Four-level
oscillator
with dipole
interactions.
Consider a four-level system with Hamiltonian H — HQ +
Ho
-Ei 0 0 0
0 0 -Eh 0 0 E2 0 0 258
0 0 0 Ex
f{t)Hi,
Limitations on Quantum Control 2111
and Orfi 0 0 di 0 d2 0 0 d 2 0 -di 0 0 -di 0 Note that \HQ and \H\ satisfy f x = - c,
xTJ + Jx = 0
(5)
for 0 0 0_-l
0 01' 0 10 100 0 0 0.
where J is unitarily equivalent to 0
IN/2
.—IN/2
0
(6)
which is a defining relation for sp(N/2). Consider an initial state of the form po = ix + OIN,
(7)
where x satisfies (5), it can only evolve into states pi = \y + aIN,
(8)
where y satisfies (5), under the action of a unitary evolution operator in exponential image of L. Hence, any target state that is not of the form (8) is not accessible from the initial state (7). Note that the initial state
Po
"0.35 0 0 0
0 0 0.30 0 0 0.20 0 0
0 " 0 0 0.15.
is of the form po = x + 0.25/4 and1 that that
IX = 1
0.10 0 0 0
0 0 0 0 0.05 0 0 --0.05 0 0 -0.10 0
satisfies (5). Consider the target state state
Pi
"0.30 0 0 0
0 0 0.35 0 0 0.20 0 0 259
0 " 0 0 0.15.
2112 A. I. Solomon & S. G. Schirmer
which is clearly kinematically admissible since
0 100] "0 1 0 0' 1000 1000 0 0 1 0 Po 0010 0001 0001. but note that pi=y
f
+ 0.25/4 and '0.05 0 0 0 " . _. 0 0.10 0 0 12/_1 0 0 -0.05 0 . 0 0 0 -0.10.
does not satisfy (5). Hence, p\ is not dynamically accessible from po for this system. Given a target state p\ that is not dynamically accessible from an initial state Po, we can easily construct observables whose kinematical upper bound for its expectation value can not be reached dynamically. Simply consider A = p\. The expectation value of A assumes its kinematical maximum only when the system is in state pi. Since p\ is not reachable, the kinematical upper bound is not dynamically realizable. 4. Conclusions In this short introduction to Quantum Control theory, we have described the goals of the subject briefly, and then illustrated the limitations by generic examples where complete control is not possible. The tools we have used are, in the main, those of classical Lie group theory. Theoretical problems that remain to be tackled include to what extent these non-controllable systems can, in fact, be controlled; and, of course, the paramount problem of implementing these controls in practice. Acknowledgements It gives us great pleasure to acknowledge many conversations with colleagues, including Drs. Hongchen Fu and Gabriel Turinici. A.I.S would also like to thank the Laboratoire de Physique Theorique des Liquides, of the University of Paris VI, for hospitality during the writing of this note. References 1. H. Fu, S. G. Schirmer, and A. I. Solomon. Complete controllability of finite-level quantum systems. J. Phys. A 34, 1679 (2001) 2. S. G. Schirmer, H. Fu, and A. I. Solomon. Complete controllability of quantum systems. Phys. Rev. A 63, 063410 (2001) 3. D. Montgomery and H. Samelson, Transformation groups on spheres. Ann. Math. 44(3), 454 (1943)
260
Internationa] Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2113-2120 © World Scientific Publishing Company
RIGOROUS RESULTS O N T H E STRONGLY CORRELATED ELECTRON SYSTEMS B Y T H E SPIN-REFLECTION-POSITIVITY METHOD *
GUANG-SHAN TIAN Department of Physics, Peking University Beijing 100871, The People's Republic of China
Received 28 October 2001 In this talk, we shall briefly review some results on the strongly correlated electron systems, derived recently by applying Lieb's spin-reflection-positivity method. To explain the basic ideas of this method to a wide audience, we emphasize t h e important role played by Marshall's rule in studying t h e many-body systems Keywords: Strongly Correlated Electron Systems, Marshall's Rule, Spin-reflection-positivity Method
In the past several decades, the strongly correlated electron systems attract many physicists' interest. In particular, interplay between the itinerant magnetic orderings and the quantum transport properties of these systems is a main focus of the current research. To get insight into the strongly correlated electron materials, various models have been introduced and investigated. The best known examples are the Hubbard model, 1 the periodic Anderson model, 2 and the Kondo lattice model.3 As a concrete example, let us consider the Hubbard model. On a lattice A with N\ sites, the Hamiltonian of the Hubbard model has the following form
Jfff =
- * E E (^+dldi°)
+U
J2 (ftit -1) (% -\)-PN,
&
ieA ^
'
^
(i)
'
where c\a (cja) denotes the fermion creation (annihilation) operator which creates (annihilates) an electron of spin a at lattice site i. < ij > is a pair of lattice sites. t > 0 and U > 0 are parameters representing the hopping energy and the onsite Coulomb repulsion of electrons, respectively. In the following, we shall assume that, in terms of the Hamiltonian, lattice A is bipartite. In other words, it can be separated into two sublattices A and B, and electrons hops only from a site i in one sublattice to a site j in another sublattice. 'Dedicated to Professor F. Y. Wu's 70th birthday. 261
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Tian
These models enjoy some symmetries, which can be exploited to simplify their analysis. For instance, the Hubbard Hamiltonian HH commutes with the total particle number operator N = Xa(^it+"u)- Therefore, the Hilbert space of this model can be divided into numerous subspaces (V(N)}. Each of them is characterized by a conserved particle number JV. HH also commutes with the following spin operators
igA
ieA
i€A
2
Consequently, both S and Sz are good quantum numbers. Furthermore, when /x = 0, the ground state of the Hubbard Hamiltonian in the half-filled subspace with N = N\ is actually its global ground state. In this case, the Hubbard Hamiltonian has another symmetry: The pseudospin spin symmetry. More precisely, HH commutes with operators J+ = '%2 e (i)Cit g U' ^ " = Y. eCOcuCi-r, Jz = 2 5 Z ( « n + "14. - 1), i6A
igA
(3)
i6A
where e(i) = 1 for i £ A and e(i) = — 1 for i £ B. In literature, these operators are called the pseudospin operators. 4 In one dimension, the Hubbard model can be exactly solved by applying the Bethe ansatz. 5 However, in higher dimensions, this approach fails. Instead, various approximate analytical techniques were introduced and developed. Most of them are based on the mean-field theories.6 By applying effectively these methods, many interesting results on the strongly correlated electron systems have been derived. On the other hand, if it is possible, one would naturally like to re-establish some of these conclusions on a more rigorous basis. It will deepen our understanding on the electronic correlations in these models. In a seminal paper published in 1989, Lieb introduced a powerful method, the spin-reflection-positivity technique, to investigate the Hubbard Hamiltonian with an even number of electrons. 7 With this method, Lieb proved that the ground state of the Hubbard Hamiltonian at half-filling is nondegenerate and has total spin 5 = (1/2)\NA — NB\, where NA and NB are the numbers of the sites in sublattice A and B, respectively. Later, this technique was also applied to both the periodic Anderson model and the Kondo lattice model. 8 ' 9 Similar conclusions were established. Later, we applied this method to investigating the spin and the off-diagonal correlation functions, as well as the excitation gaps of these strongly correlated electron models. In a series of papers, we proved that • The on-site pairing correlation function of the negative-C/ Hubbard model is nonnegative. 10 More precisely, for any pair of lattice sites h and k, inequality
<*o(-C0 I 4 e h 4 ^ c k t I *o(-EO> > 0
(4)
holds true for the ground state of the negative-?/ Hubbard model in V(2M). This inequality confirms that the Bose-Einstein condensation in the negative-?/ 262
Rigorous Results on the Strongly Correlated Electron
Systems
2115
Hubbard model occurs at zero-momentum. • At half-filling, the spin correlations in the ground states of the periodic Anderson model and the Kondo lattice model are antiferromagnetic. 11 Very recently, by extending the spin-reflection-positivity method to the case of nonzero temperature, we also proved that, at any T ^ 0, the antiferromagnetic spin correlation in these models is dominant. 12 • On some special lattices subject to condition \NA — NB\ = 0(NA), such as the example shown in Fig. 1, the ground state of the Hubbard model has both the antiferromagnetic and ferromagnetic long-range orders. In other words, it is a ferrimagnet.13 • Define A q p = E0(NA + 1) + E0(NA - 1) - 2E0(NA) and As = E0(NA, S = 1) — EQ(NA, S = 0) to be the quasi-particle gap and the spin excitation gap of the Hubbard model, respectively. Then, relation A q p > Ag holds true. Similar inequalities were also proven for the periodic Anderson model and the Kondo lattice model.14 • Define A c = E0(J = 1)-E0(J = 0) = E0(NA+2)-E0(NA) to be the charged gap of these strongly correlated electron models, then inequality A c > A s holds. 15
Fig. 1.
The lattice structure of organic conjugated polymers.
Due to the page limit of this paper, it is impossible to discuss the spin-reflectionpositivity method and its applications in details. In the following, we shall briefly explain the basic ideas of it. As a matter of fact, this method is closely related to a very simple but important observation: After a proper unitary transformation, the ground state of a realistic quantum many-body Hamiltonian, in general, satisfies Marshall's rule,16 in a sense. To begin with, let us first consider a simple quantum mechanical system: One particle moving in a one-dimensional well, as shown in Fig. 2. The ground state ^o(x) of this system satisfies the Schrodinger equation
h2 d2ty0(x)
. .T , .
_
T
. ,
(5)
When v(x) — 0, $o{x) can be explicitly solved and we have ^o(x) = y/2/Rsm(Trx/R) > 0 for any 0 < x < R. This is Marshall's rule for <3>o in this special case. A direct corollary of this rule is that the ground state ^o(^) is nondegenerate. 263
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G.-S.
Tian
Fig. 2.
T h e potential function of a one-dimensional quantum well.
To show that ^?o(%) also satisfies the Marshall's rule when v(x) ^ 0, we notice that tyo(x) is a ground-state solution of Eq. (5), if and only if it is also a minimizing function of the following energy functional
n2 fRm{x)
™=LL
\
'*
dx
x)dx
(6)
Jo
in some function space -ff1(0, R), requiring that both l^l 2 and \dif)/dx\2 are integrable over (0, R). Assume that ^o(x) has indefinite sign in the interval, as shown in Fig. 3. Then, i
\ / /// // //
iV
k
l\
1/ \\
/1 //
\\ \\
A
/\ \\
i/
V=oo
« A /
1 \ l\ // / i l l // / / 11 1 1 1/
J—\—/-U-l / Fig. 3.
IW
V=o°
•
0
R
X
T h e ground state wave functions *o(^) and |*o|(z)-
we construct a new function |\Po|(2:) by taking the absolute value of $o(x). We notice that replacement of ^Q{X) with l^olOc) does not change the value of the second term in Eq. (6). On the other hand, it can be shown
/' Jo
dV0(x) dx
dx >
f
d | *o | (x) dx
dx
(7)
Jo (See the appendix of Ref. 17). Consequently, we have
min £(i/>) = £ ( * „ ) > £(\ *o I)
(8)
In other words, | \&o | (x) must be also a minimizing function of energy functional (6) and hence, a ground state solution of Eq. (5). 264
Rigorous Results on the Strongly Correlated Electron Systems
2117
Next, we show that the ground state wave function ^o{x) has no zero point in (0, R). If this is not true, then, there must be, at least, one point XQ € (0, R) such that |^o|(a;o) — 0. However, as a nonnegative solution of a second order elliptical differential equation, |\&o|(a;) satisfies the so-called Harnack theorem, which tells us that, on any open interval (a, b) C (0, R), there is a constant C(aj;,) such that max | *o I (&) < C ( o , b) min x£(a, b)
| * 0 I (x)
(9)
xE(a, b)
holds true. 18 Now, we take an open interval containing xo- Harnack theorem implies that l^oK^) = 0 in this interval. Repeating this argument an appropriate number of times, we find that | ^o | must be identically zero in (0, R). This is certainly absurd. Therefore, we reach the conclusion that ^o(x) = |*o|(a0 > 0 in (0, R) and is nondegenerate. The same ideas can be also applied to study the strongly correlated electron systems. First, let us consider a simple model: The antiferromagnetic Heisenberg model on a bipartite lattice. The Hamiltonian of this model is of the following form
HAF = £ JySi • 4 = 1 J2 J« ( ^ + 3 - + Si-Sj+) + Y, JaS-A <«>
(10)
with Jy > 0. Si is the spin operator at site i. The lattice A is assumed to be bipartite and connected by these coupling constants. Since Hamiltonian (10) commutes with Sz = ^2iS-lz, Sz is a good quantum number. A natural basis of vectors in subspace V(SZ = M) is given by Xa(M) = | m i , m 2 , •••, mNA)
(11)
where ra\ represents the eigenvalue of S-lz at site i and they are subject to the condition m\ + W2 H h TUNA = M. In terms of this basis, the ground state wave function ^o(M) of H\F can be written as *o(M) = ^ C Q X a ( M )
(12)
a
The sum is over all the possible configurations {xa{M)}. However, due to the positive sign of the coupling constants {Jy}, it is difficult to uncover directly the sign rule satisfied by {Ca}. To remedy this problem, one needs to introduce a unitary transformation U\ = exp [i^^Z^B S-lz 1, which rotates each spin in sublattice B by an angle 7r about its z-axis. 19 Under this transformation, HAF is mapped onto HAF = utHxeUi
- Yl ( - - V 2 ) (Si+Si-
+ 5,_5,+) + ] T J u S „ S J z
(13)
Notice that the coupling constants in the spin-flipping terms of HAF have negative signs. For HAF, we are able to show that the expansion coefficients {Ca} of its ground state ^o satisfy the Marshall's rule Ca > 0. First, by following the aforementioned 265
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G.-S.
Tian
steps, we re-write the ground state energy EQ as an expectation value of HAF in its ground state ^o a n d observe that, for any pair of indices a and a', (x«(M)
| ( s , + S j _ + 5 i _ 5 j + ) | Xa'{M))
>0
(14)
holds true. Consequently, the state |\l/o|, which is constructed by replacing each Ca with \Ca\ in Eq. (12), has a lower energy than \to- Therefore, it is also a ground state. Next, we consider the Schrodinger equation of |\Po|- By inequality (14), it can be shown that, if Ca = 0 for some index a, then any vector \a', which is related to Xa by a spin-flipping exchange, must have a zero coefficient in the expansion of l^o |- On the other hand, since the lattice is connected by the coupling constants {Jjj}, any vector X/3 c a n be reached from Xa by a finite number of spin-flipping exchanges. Therefore, by repeating the above process an appropriate number of times, we reach the conclusion that all the expansion coefficients are zero. That is impossible. As usual, the Marshall's rule satisfied by {Ca} implies that the ground state ^o(M) of i?AF is nondegenerate in subspace V(M). On the other hand, since ETAF is unitarily equivalent to the original antiferromagnetic Heisenberg Hamiltonian, we conclude that the ground state ^o(M) of the antiferromagnetic Heisenberg Hamiltonian HAF hi V(M) is also nondegenerate. Finally, we consider the Hubbard model. As explained above, we first introduce a unitary transformation which maps the original positive-?/ Hubbard model into a negative-?/ Hamiltonian. This can be achieved by the so-called partial particle-hole transformation U2,4 which is defined by ulctxUi = e{\)c\v
^ 2 t c u t> 2 = c u .
(15)
Under this transformation, the positive-E/ Hubbard Hamiltonian at half-filling (with Lb = 0) is mapped to
HH(-U) = -t £ £ (4,q* + i ^ ) - U £ (n,t - \) (nu - i) . (16)
ieA ^
'
^
'
When N, the number of electrons in the system, equals an even integer 2M, the ground state wave function ^o(2M) of HH(—U) can be written as
*o(2M) = X V
a
^®^
(17)
a, /3
where {i/^} are configurations of electrons of spin a defined by
V£ = < A r •••<,* I °>
(18)
In Eq. (18), (ii, 12, • • •, \M) denotes the lattice sites occupied by fermions of spin a. I 0) is the vacuum state. The total number of these configurations is Cff . In the Hubbard model, electrons are itinerant. Therefore, we have the so-called fermion sign problem. Consequently, the coefficients {WQ;g} do not satisfy the simple 266
Rigorous Results on the Strongly Correlated Electron Systems
2119
Marshall's rule. However, Lieb proved that, if one takes index a for row index and (3 for column index and writes the coefficients { W ^ } into a matrix W (with I r W ^ W = 1), then this matrix is, in fact, a positive-definite matrix. It is the generalized Marshall's rule for the ground states of the negative- U Hubbard model. To prove this fact, we follow the above procedure and rewrite EQ{—U) as the expectation of HH(-U) in ^0(2M). A little algebra yields Eo{-U) = 2Tr(TWW) - U ^
THV^MVWVi)
(19)
i€A
where T is the matrix of the hopping term of HH(—U) and N\ is the matrix of operator h\ — 1/2. Since matrix W is Hermitian, we can find a unitary matrix V, which diagonalizes it. Let {wm} be the eigenvalues of W and {| m)} be the column vectors of the diagonalizing matrix V. Then, EQ{—U) can be further reduced to E0(-U) = 2^(m
| f | m)w2m - U ^ Y ,
m
w
^n\{n \ N, \ m)\2
(20)
iGA m, n
Obviously, if we replace {wm} with their absolute values, the summations on the right hand side of Eq. (20) becomes less. In other words, by replacing the coefficient matrix W of * 0 (2M) with | W |, we obtain a new wave function | * 0 | ( 2 M ) , which has a lower energy than the ground state $o(2M). Therefore, it must be also a ground state of -fl#(—U). Furthermore, its coefficient matrix | W | is a semipositive definite matrix. By substituting |* 0 |(2M) into the Schrodinger equation HH(-U)\^0\ = Uol^ol and noticing that the lattice A is connected by electron hopping, we can further show that, if one eigenvalue wm = 0, then all the eigenvalues of W are equal to zero. This is impossible. Therefore, W is actually a positive definite matrix and the ground state $o(2M) is nondegenerate. Since HH(-U) is unitarily equivalent to HH{U) at half-filling, the ground state of the latter Hamiltonian is also nondegenerate. With the positive definiteness of the coefficient matrix W of $o(2M), inequality (4) can be proven as a direct corollary. Similarly, other results listed above are also based on this fact, although their proofs require a little more effort. In summary, the spin-reflection-positivity method reveals that the ground states of some strongly correlated electron systems satisfy the Marshall's rule in a more sophisticated manner. In return, this rule enables us to establish several important qualitative properties on the spin and superconducting correlations as well as the excitation gaps in these systems. Acknowledgement s I would like to thank Prof. Mo-Lin Ge for inviting me to the APCTP-Nankai Symposium. This work is partially supported by the Chinese National Science Foundation under grant No. 19874004. 267
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References 1. J. Hubbard, Proc. Roy. Soc. London A 2 7 6 , 238 (1963); M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963); J. Kanamori, Pro. Theor. Phys. 30, 275 (1963). 2. For a review, see, for example, P. A. Lee, T. M. Rice, J. W. Serene, L. J. Sham, and J. W. Wilkins, Comments Condens. Matter Phys. 12, 99 (1986). 3. For a review, see, for example, G. Aeppli and Z. Fisk, Comments Condens. Matter Phys. 16, 155 (1992); H. Tsunetsugu, M. Sigrist, and K. Ueda, Rev. Mod. Phys. 69, 809 (1997). 4. C. N. Yang and S. C. Zhang, Mod. Phys. Lett. B4, 759 (1990). 5. E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968). 6. For a detailed review on these methods, see E. Fradkin, Field Theories of Condensed Matter Systems (Addison-Wesley, Redwood City, California, 1991). 7. E. H. Lieb, Phys. Rev. Lett. 62, 1201 (1989). 8. K. Ueda, H. Tsunetsugu, and M. Sigrist, Phys. Rev. Lett. 68, 1030 (1992). 9. T. Yanagisawa and Y. Shimoi, Phys. Rev. Lett. 74, 4939 (1995); H. Tsunetsugu, Phys. Rev. B55, 3042 (1997). 10. G. S. Tian, Phys. Rev. B 4 5 , 3145 (1992). 11. G. S. Tian, Phys. Rev. B 5 0 , 6246 (1994). 12. G. S. Tian, Phys. Rev. B 6 3 , 224413 (2001). 13. S. Q. Shen, Z. M. Qiu and G. S. Tian, Phys. Rev. Lett. 72, 1280 (1994); G. S. Tian and T. H. Lin, Phys. Rev. B 5 3 , 8196 (1996). 14. G. S. Tian, Phys. Rev. B 5 8 , 7612 (1998). 15. G. S. Tian and L. H. Tang, Phys. Rev. B 6 0 , 11336 (1999); J. G. Wang and G. S. Tian, Commun. Theor. Phys. 34, 21 (2000). 16. W. Marshall, Proc. R. Soc. London A 2 3 2 , 48 (1955). 17. J. K. Freericks and E. H. Lieb, Phys. Rev. B 5 1 , 2818 (1995). 18. N. S. Trudinger, Comm. Pure Appl. Math. 20, 721 (1967). 19. E. Lieb and D. Mattis, J. Math. Phys. 3, 749 (1962).
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International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2121-2127 © World Scientific Publishing Company
AN ALGEBRAIC APPROACH TO THE EIGENSTATES OF THE CALOGERO MODEL HIDEAKI UJINO Gunma National College of Technology, 580 Toriba-machi Maebashi-shi, Gunma-ken 371-8530, Japan
Received 9 November 2001 An algebraic treatment of t h e eigenstates of the (Ajv-i-)Calogero model is presented, which provides an algebraic construction of the nonsymmetric orthogonal eigenvectors, symmetrization, antisymmetrization and calculation of square norms in a unified way.
1. T h e Calogero Model In 1990's, one-dimensional quantum integrable systems with inverse-square longrange interactions 1-3 attracted renewed interests of mathematicians and physicists since their relationships with the theory of multivariable orthogonal polynomials 4 ' 5 are recognized. The (A/v-i-)Calogero model1 with distinguishable particles 6
where (Kjkf){-
• • ,xjt
• • • ,xk, • • •) = / ( • • • ,xk, •• -,Xj,- • •), j,k
G {1,2, • • •,JV}, is
known to have the nonsymmetric multivariable Hermite polynomial as the polynomial part of the joint eigenvector of the conserved operators.7~9 We introduce a transformed Hamiltonian whose eigenvectors are polynomials, %W := ( ^ ( a O r o
1
^ - EW) O W(x),
(2)
where the reference state and its eigenvalue are
4A\x) = n \x, -x fc rex P (-^ f; xi), l<j
EM =
m=l
^uN(Na+(l-a)).
We shall deal with the eigenvectors of the Hamiltonian (2) in C[x], the polynomial ring with N variables over C, while the eigenstates for the original Hamiltonian (1) 269
2122
H. Ujino
is in C[z]4 A ) := {f{x)<j>{^\x)\f G C[x]}. The Hamiltonian (2) is Hermitian with respect to the inner product on C[a;], oo
N
/
"[[Axi\4>(kA\x)\2f{x)g{x), f,geC[x], (3) where f(x) denotes the complex conjugate of f(x). The inner product is induced from the natural inner product on C[a;]0g '. The reference state corresponds to the weight function in the inner product (•, -}(A)The commutative conserved operators for the Hamiltonian are known to be the Cherednik operators. 10 To show this, we have to introduce the Dunkl operators, 11
and the creation-like and annihilation-like operators, M)\ ._ _ J_V{A) (A) _ J _ V ( A ) *' -Xl 2u,V< ' ai ~ 2 u l in G End(C[a:]), where the superscript t on any operator denotes its Hermitian conjugate with respect to the inner product (3). Prom these operators, a set of Hermitian and commutative differential operators, dj ' G End(C[x]), [dj ,d\. '] = 0, is constructed by N
k=j+l 10 12
which we call the Cherednik operators. '
The Hamiltonian (2) can be expressed
HM=UJ:(4A)-\a(N-l)). Thus we conclude that the Cherednik operators {cf- '\j = 1,2, • • •, A''} give a set of commutative conserved operators of the Calogero model. 2. The Nonsymmetric Multivariable Hermite Polynomial The Cherednik operators define inhomogeneous multivariable polynomials as their joint polynomial eigenvectors, which are nothing but the nonsymmetric multivariable Hermite polynomials that form an orthogonal basis of the polynomial ring C[:r]. 7 - 9 ' 13 Let / := {1, 2, • • • ,N - 1} and I := {l,2,---,iV} be sets of indices and let V be an iV-dimensional real vector space with an inner product (•,•). We take an orthonormal basis of V {ej\j G / } such that (£j,£fc) = Sjk- The A^-itype root system R associated with the simple Lie algebra of type AN~I is realized as R = {ej — Sk\j,k € I,j ^ k}(c V). A root basis of R is defined by 270
An Algebraic Approach to the Eigenstates
2123
II := {otj = £j — £j+i\j G / } , whose elements are called simple roots. Let R+ be positive roots relative to LT and R- :— — i?+. The root lattice Q is denned by Q := 0 g/Zccj and positive root lattice Q+ is defined by replacing Z with Z>oA reflection on V with respect to the hyperplane that is orthogonal to a root a is expressed by sa(p) := p — {a.v,p)a, where av := i^x is a coroot corresponding to a. The yl^-i-type Weyl group is generated by {SJ :— saj\aj G II} which is isomorphic to 6JV- For each w G W, we define Rw := R+ D w; -1 i?_. We denote by £(w) the length of w G W defined by £{w) := \Rw\. When w G W is expressed as a product of simple reflections, w = Sjk •••Sj2Sj1, with k — £(w), we call it reduced. By use of the reduced expression, the set R^, is given by R>l>
=
\ajl
' Sjl
\0lJ2 ) > ' ' ' J Sjl
S
J2
' ' ' Sjl
-1 \ajl
) J•
We introduce lattices P := © j e / Z > 0 £ j and P+ := {p = YLjzi Mj£j e -^Vi > M2 > • " > Miv > 0}, whose elements are called a composition and a partition, respectively. The lattice P is ^ - s t a b l e . The degree of the composition and partition is denoted by \p\ := X^e/£*j- Let W(p) := {w(/i)|io G W } be the VF-orbit of p G P . In a VF-orbit of W(/i), there exists a unique partition p+ G P+ such that p = w(p+) G P (W G W). We define p := § £ a £ f l + « = | E j e / ( ^ - 2j + l)£j and 1N := J2jei £r We introduce the following operator SA^X := J2jei ^i^j > ^ e ^*> j G /, which relates the Cherednik operators with lattice P , so that we can deal with the eigenvalues of the Cherednik operators in terms of the lattice P . We identify the elements of the lattice P with those of the polynomial ring over C, x* := x^x^2 •••x%N G C[x}. We denote the shortest element of W such that w~'i-{p) G P+ by Wy, and define p(p) := w^p). We introduce an order X on P ,
v
{y,pGP)^\v+<^
»?W(ii+) [ ( i - ^ e Q i Z/G W{p+),
d
where the symbol < denotes the dominance order among partitions, d
v
'
(ji,v G P+) <^>/i ^ A, \p\ = \v\ and J^ffe < k=i
'
^Pk, fe=i
for all I G / . The definition of the nonsymmetric multivariable Hermite polynomial is summarized as follows. Definition 1: The (monic) nonsymmetric multivariable Hermite polynomial hp ' G # > ( < * , - j - ) s " G C[x], 2u>' or u-su. |i/|<|f*|
SA»hM
= (A, p + ap{p) + L(N
271
- l)lN)hW.
(4)
2124
H. Ujino
Since SA^X is a Hermitian operator with respect to the inner product (3) and all the simultaneous eigenspaces of the Cherednik operators {d,(A^} are onedimensional in the sense that the eigenvalues of {d,(A^} are non-degenerate, it proves that the polynomials hfj, ' are orthogonal with respect to the inner product, i.e., (hfj. ,h{> )(A) = ^.I/H^M ||2- Actually, the nonsymmetric multivariable Hermite polynomials form an orthogonal basis in C[x\. 3. The Rodrigues Formula Here we show the Rodrigues formula that generates the monic nonsymmetric multivariable Hermite polynomial. We introduce the Knop-Sahi operators {e^A\ e^A^}7'14 and the braid operators {Sj '\j G 1} defined by c ^ := a{A^KlK2
• • • Ks-u
where Kj := Kjj+i,
j G / . The operators {SJ ', e^A^} intertwine the simultaneous
eigenspaces of {d^x}.
AU»
e™ = KN^
• • • K2Kxa^B)\
flf
> :=
[Khdf\
The raising operators {A^ ' \/j, G P+} are defined by
; = (S<^)S(J,B)
... s M,fl) e (A, B )ty >
j€L
Let SW/X be defined by Sw^ := 5,-, • • • Sj2Sj1, where w^ = Sjl • • • Sj2Sj1 is one of the reduced expressions of w^. Then we can show the following relations, = A^(SA»
d(mA^
[ 4 A ) t , 4 A ) t ] = 0,
+
for M,!/ G P + ,
A A
St£>d< > = d^J'-M (*)$<£, for /i G P,
(5)
which lead to the Rodrigues formula for the monic nonsymmetric Hermite polynomial. 9 Theorem 2: The monic nonsymmetric multivariable Hermite polynomial njj, with a general composition fi G P in the W-orbit of the partition /x+ G P+ is algebraically obtained by applying the raising operator A + and the product of braid operators SwJ to h^
— 1,
hiA) = {c^)^S^A^h[A\
h^
= ltfAS
w(/1+)>
„+
e P+>
where the coefficient of the top term is expressed as
TT l r r r - 7 -v^ v ^ ^ . = TT (« v ^ + + qp) $••= n ifH-«(« .p)). <£?== n
c ^)-=
Proof: Using Eq. (5), we can confirm that /i}j given by the above formula satisfies definition 1. • 272
An Algebraic Approach to the Eigenstates
Using the square norm for h0 ,, (A) , (A), ^ o ,fto ;CA) -
2125
— 1, (2?r)f (2w)iJV(JVo+(1_0))
j-rTjl+ja) JLl r ( i + a ) '
which is proved by a certain limit of the Selberg integral, 15 and the Rodrigues formula, we can calculate the square norm of the nonsymmetric multivariable Hermite polynomial in an algebraic fashion.9 Theorem 3: The square norms of the nonsymmetric multivariable Hermite polynomial hi ' with a general composition fj, 6 W(fj,+), fi+ € P+ is given by 9
=
^Wi^d-)HW j-r
11 (/3V ) M + + a^-a 2 ll r K + + a ^-^) + 1)
r ( ( a v , n+ + ap) + 1 + a ) r ( ( a v , /x+ + ap) + 1 - a)
Proof: The above formula can be verified by use of relations among raising and braid operators and evaluation of the eigenvalues of the Cherednik operators. • 4. Symmetrization and Antisymmetrization The nonsymmetric multivariable Hermite polynomials with compositions fi in the same W-orbit of the partition /j,+ share the same eigenvalue of the Hamiltonian (2), tfA)hW=u>\vL+\hW,
forMGW(M+),
/i+SP+.
More generally, the polynomial with compositions /j, € W(fi+) share the same eigenvalue of an arbitrary symmetric polynomial, e.g., any of the power sums, of the Cherednik operators. Thus any linear combinations of hjj, ', n G W(fi+), fi+ € P+ are joint eigenvectors of the Calogero Hamiltonian (2) and its higher-order conserved operators. Among all such linear combinations, we consider eigenvectors of the Calogero Hamiltonian (2) in VF-symmetric and PF-antisymmetric polynomial rings over C, C^] 1 * 1 ^. In our formulation, we do not use symmetrizer or anti-symmetrizer 7 which makes the coefficients of the top terms differ from unity. We introduce a sublattice of P+ by P+ +5 := {/J + <5|/Z e P+}, S := Y,J<EI(-^ ~J)£j t o describe the antisymmetric eigenvectors. Theorem 4: 9 Let # £ + ) + for /i+ € P+ and H^~ for n+ eP++6be the following linear combination of the nonsymmetric multivariable Hermite polynomials with compositions \i £ W(n+),
*#* = £ *£W, 273
(g)
2126
H. Ujino
whose coefficients are
Then the polynomials are in symmetric or antisymmetric polynomial rings over C, i.e., H^J G C[x]±w. We call them the symmetric and antisymmetric multivariable Hermite polynomials, respectively. Proof: Requiring the linear combination of the nonsymmetric polynomial (6) to satisfy KjHfr ' = ±HJi , b( + + = 1, we obtain the coefficients b + as shown in Eq.(7). D The symmetric and antisymmetric multivariable Hermite polynomials are identified by the polynomial parts of the eigenstates for all the conserved operators of the (j4iv_i-)Calogero model with indistinguishable (bosonic or fermionic) particles,1' 17 - 18 N
f)2
i
*
^M^EHS+^+JX: 3
J=l
_ O2' .^ t t
j,k=l
which is obtained by restricting the operand of the Hamiltonian (1) to the space C[*]±^)>^)|CM±lir^,=^)±(a). Prom the square norms of the nonsymmetric polynomials (/ij, %/iJ, )(A) and the coefficients bu+u , we can evaluate the square norms of the (anti-)symmetric multivariable Hermite polynomials. To prove the formula of the square norms, we need the following lemma, 16 Lemma 5: For p. G P+, we have an identity, v
(a ,p, H+ [IX t aap) f f i Ta fa
E/ N 1™ (a ,u + ap)±a 1 / v,, + \ + „. -pr
=
w
n
nn
(av,p + ap) *•*• (av,a + ap) ± a ' 11 j-r T-r
The lemma is proved by use of an expression of the Poincare polynomials.19 Theorem 6: 9 Let H{^)+ for p+ G P+ and H(^]~ for p+ G P+ + S. The square norms of the (anti-)symmetric multivariable Hermite polynomials are given by
T-r r ( ( a v , p + ap) + 1 T a ) r ( ( a v , / j + ap) ± a) J |
T((a\p + ap) + l)T((a^p + ap))
Proof: The proof is straightforward from theorems 3 and 4, and lemma 5. 274
•
An Algebraic Approach to the Eigenstates
2127
5. C o n c l u d i n g R e m a r k s We have presented the Rodrigues formula for t h e monic nonsymmetric multivariable Hermite polynomial which gives t h e n o n s y m m e t r i c orthogonal eigenfunctions of t h e ( J 4jv_i-)Calogero model with distinguishable particles. T h r o u g h (anti)symmetrization, we have constructed t h e (anti-)symmetric Hermite polynomials t h a t give t h e polynomial p a r t s of the eigenfunctions of t h e (^4jv-i-)Calogero model with distinguishable particles. T h e square n o r m s of t h e above three cases are calculated in a n algebraic manner. Our formulation in this work is also applicable t o the i?;v-Calogero models with distinguishable or indistinguishable particles a n d results in the Rodrigues formula for the monic n o n s y m m e t r i c multivariable Laguerre polynomial a n d square norms of the nonsymmetric a n d (anti-)symmetric multivariable Laguerre polynomials. 9 Acknowledgements I would like to express my sincere g r a t i t u d e t o Professor M. L. Ge for kindly inviting me to t h e symposium and to all t h e staffs a t Nankai University for their warm hospitality. I am grateful to Dr. A. Nishino for fruitful collaboration. I appreciate the Grant-in-Aid for Encouragement of Young Scientists (No. 12750250) presented by t h e J a p a n Society for the Promotion of Science for financial s u p p o r t . References 1. 2. 3. 4. 5. 6. 7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
F. Calogero, J. Math. Phys. 12 419 (1971). B. Sutherland, Phys. Rev. A 4 2019 (1971). M. A. Olshanetsky and A. M. Perelomov, Phys. Rep. 94 313 (1983). I. G. Macdonald, Symmetric Functions and Hall Polynomials 2nd. ed. (Oxford: Clarendon Press, 1995). C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables (Cambridge: Cambridge University Press, 2001). A. P. Polychronakos, Phys. Rev. Lett. 69 703 (1992). T. H. Baker and P. J. Forrester, Nucl. Phys. B 492 682 (1997). H. Ujino and A. Nishino, Special Functions, Proc. Int. Workshop (Hong Kong, June, 1999) eds. C. F. Dunkl, M. Ismail and R. Wong, (Singapore: World Scientific, 2000) p. 394. A. Nishino and H. Ujino, J. Phys. A: Math. Gen. 34 4733 (2001). I. Cherednik, Inv. Math. 106 411 (1991). C. F. Dunkl, Thins. Amer. Math. Soc. 311 167 (1989). S. Kakei, J. Math. Phys. 39 4993 (1998). C. F. Dunkl, Commun. Math. Phys. 197 451 (1998). F. Knop and S. Sahi, Inv. Math. 128 9 (1997). M. L. Mehta, Random Matrices 2nd. ed. (San Diego: Academic, 1991). A. Nishino and M. Wadati, J. Phys. A: Math. Gen. 33 3795 (2000). H. Ujino and M. Wadati, J. Phys. Soc. Japan 6 5 2423 (1996). H. Ujino and M. Wadati, J. Phys. Soc. Japan 6 6 395 (1997). J. E. Humphreys, Reflection Groups and Coxeter Groups, (Cambridge: Cambridge University Press, 1990).
275
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2129-2136 © World Scientific Publishing Company
ONSAGER'S ALGEBRA A N D PARTIALLY O R T H O G O N A L POLYNOMIALS
G. VON GEHLEN* Physikalisches Institut der Universitdt Bonn Nussallee 12, D-53115 Bonn, Germany
Received 20 January The energy eigenvalues of the superintegrable chiral Potts model are determined by t h e zeros of special polynomials which define finite representations of Onsager's algebra. T h e polynomials determining the low-sector eigenvalues have been given by Baxter in 1988. In the Z3—case they satisfy 4-term recursion relations and so cannot form orthogonal sequences. However, we show that they are closely related to Jacobi polynomials and satisfy a special "partial orthogonality" with respect to a Jacobi weight function. PACS: 02.30.1, 75.10.J, 68.35.R Keywords: Chiral Potts model, Integrable quantum chains, Onsager's algebra
1. Introduction F.Y.Wu and Y.K.Wang1 were the first to consider the Potts model with chiral interaction terms. Their interest in this generalization arose from duality considerations, but the idea proved to be very fruitful in many respects: Ostlund 2 and Huse 3 proposed the chiral Potts (CP) model for phenomenological applications: it allows to describe incommensurate phases using nearest neighbor interactions only. We give a few references4-7 from which the subsequent development can be traced, and turn directly to the superintegrable chiral ZN Potts quantum chain. 8 This is a particularly interesting model, because it provides some of the rare representations known for Onsager's algebra9 and in this sense generalizes the Ising quantum chain (for N = 2 it is the Ising model). Integrability by Onsager's algebra entails that all eigenvalues of the hamiltonian are determined by the zeros of certain polynomials, which for the chiral Potts model were first derived by Baxter. 10 Although the definition of Baxter's polynomials looks very simple, the properties of these polynomials turn out to be quite non-trivial and interesting. 11 The main part of this note deals with the properties of these polynomials. They satisfy N + 1-term recursion relations, therefore for N > 2 they cannot form orthogonal sequences. However, as found recently, 11 several properties which characterize orthogonal polynomials * e-mail: [email protected]. uni-bonn.de 277
2130
G. von Gehlen
are almost true for Baxter's polynomials (e.g. the zero separation property is true except for one extreme zero). We first recall the definitions of the superintegrable CP-hamiltonian and Onsager's algebra and then, following B.Davies, 12 we sketch how the formula for the energy eigenvalues emerges. We consider Baxter's polynomials and their recursion relations. Equivalent polynomials with their zeros in (—1,-1-1) for N = 3 are written in terms of a determinant. Their expansion in terms of Jacobi polynomials gives the surprising result that many of the expansion coefficients vanish, leading to the notion of "partial orthogonality". The hamiltonian defining the Zjv-superintegrable chiral Potts quantum chain 8 ' 8 is:
Here u — e 2 " ' ^ and Zj and Xj are Zjv-spin operators acting in the vector spaces C^ at the sites j = 1, 2 , . . . , L (L is the chain length). The operators obey ZiXj = Xj ZiUjSi<j; Zj* = X^ = 1 and we assume XL+I = X\ (periodic b.c). A convenient representation is (-Xj-)j,m = <Sz,m+i mod N and (Zj)iiTn — <Jjjmwm. For N > 3 the complex coefficients make the chain hamiltonian parity non-invariant. For TV = 2 we get the Ising quantum chain. For fixed JV there is only one parameter, the temperature variable k. Incommensurate phases arise due to ground state level crossings. H commutes with the Z^-charge Q = rjj=i Zj. We write the eigenvalues of Q as UJQ. Q = 0 , 1 , . . . , JV — 1 labels the charge sectors of %. We split H into two operators writing H^ = —^N(A0 + kAi). A remarkable property of H is that A0 and Ai satisfy 8 the Dolan-Grady 13 relations [i4o,[4>,[4),j4i]]] = 16 [AcAx];
[Au [Au [Au A0}}} = 16^,
A0],
which are the conditions 14 for A0 and A\ to generate Onsager's algebra A, which is formed9 from elements Am, Gi, m € Z, / G N, l> m, satisfying [Ai,Am] = 4Gi-m;
[GuAm]
= 2Am+i-2Am-i;
[GhGm]=0.
(2)
From (2), there is a set of commuting operators which includes %: Qm = | (Am + A-m + k(Am+1
+ A-m+i));
[Qi,Qm]
= 0;
Qo = 'H-
To obtain finite dimensional representations of A we require the Am (and analogously the G/) 1 2 ' 1 5 to satisfy a finite difference equation: ^2^-_n cik Ak-i = 0. This is solved introducing the polynomial (the main object of the present paper): n
T(z) = J2 a" zk+n k=—n 278
(3)
Onsager's Algebra and Partially Orthogonal Polynomials
2131
(from A the ak are either even or odd in k). Now the Am and Gm can be expressed in terms of the zeros Zj of T(z) and the set of operators Ef, Hf. A
m = 2 £ (
G
- = E (*™ - *7m) Hi-
(4)
Prom A these operators obey sZ(2, C)-commutation rules: [Ef, E^\
= Sjk Hk;
[H^ Ef] = ±2 Sjk
Ef.
So A is isomorphic to a subalgebra of the loop algebra of a sum of sl(2, C) algebras. Prom the first of eqs.(4) we can express "H in terms of the Zj and the operators Ef. Writing Ef = JXij ± iJyj, then in a representation Z(n,s) characterized by the polynomial zeros z\,... ,zn and a spin-s representation J^' of all the Jj, we get: (A0 + hA1)z{n,s)
= 2 X {(2 + k (zj + z?))j$.
= where J'xj
4 ^ 1
+
2 f c c
J +
+ i(Zj - zj1)
jW}
^ t !
is a rotated S'[/(2)-operator, and Cj = cosOj = \(ZJ + z~x).
For the CP-hamiltonians (1) the spin representation turns out to be s = \. Accordingly, all eigenvalues of (1) have the form # W = -N
( a + bk + 2 £™ = 1 mj
v/l+2fccos%
+ A;2) ,
m, = ±\.
(5)
a and b are non-zero if the trace of Ao and A\ is non-zero. 2. Baxter's polynomials No direct way is known to find the polynomials T(z) from the hamiltonian %. However, the invention of the two-dimensional integrable CP model, 16 ' 17 which contains W a s a special logarithmic derivative, and functional relations for its transfer matrix have enabled Baxter 10 to obtain the polynomials for the simplest sector of H, which at high-temperatures contains the ground state (the polynomials corresponding to all other sectors have been obtained subsequently in 1 8 - 2 0 ). Here we shall consider only the simplest case. Baxter 10 finds that in terms of the variable t or s = tN = (c — l ) / ( c + 1) (recall c = cos# of (5)) these polynomials take the form P
QL)(*) = j? E ( i T - Q
bJt)-**;
aQ,L = (N- 1)(L + Q) mod N. (6)
Here Q denotes the Zjy-charge sector. For Z3 eq.(6) is, written more explicitly: • i-\
P£\S)
-t~aQ,L
= !
_
{{f + t + 1)L + OjQ(? + UjH + W)h + U-Q(t2 + Ljt + U>2)L) . 279
2132
G. von Gehlen
Due to their Zjy-invariance t -> ut, the PQ' depend only on s = tN. The degree of the PQ '(S) in the variable s isfe^g= [((N — 1)L — Q)/N] where [a;] denotes the integer part of x. Considering sequences of these polynomials for fixed Q and L € N, we notice that the dimensionsft^Qdo not always increase by one when increasing L by one: at every Nth step the dimension stays the same: e.g. the dimensions of the PQ-0 for L mod N — 0 and L mod N = 1 coincide, see Table 1. The polynomials (6) have their zeros all on the negative real s-axis: % is hermitian and so in (5) we must have — 1 < Cj < +1 which means negative sj. We will prefer to deal mostly with equivalent polynomials in the variable c, defining
nW(c) = (c + i ) ^ p W ( s = f = i ) .
(?)
Our main concern in this paper is to learn about the properties of the IIQ ' (c) or Pk (S), e.g. whether these can be arranged into orthogonal sequences etc. We will find that the ITQ (C) are polynomials with quite remarkable properties. A number of special features of the UQ ' have been discussed recently. 11 Here we give some more detailed results for the Z3-case. As the recursion relations for the UQ ' contain a lot of information, we now show how to obtain these.
3. Recursion relations We start with the observation that the coefficients of the PQ \S) can be obtained from the expansion of (1 + 1 +12 +... + tN_1)L, simply by taking every N—th term of the expansion, starting with the coefficient of t(-N~1^L~Q. More precisely, we claim that we can define the PQ by the decomposition
^+t
+
e + ,,.+tN^)Lj}^\L V
^
N
^t.L,QpQL){s)
= J
(g)
Q=0
demanding the PQ to depend on s = tN only. Proof: Insert (6) into (8) to get: /
n
N \L l1 - tJ.N \
,.
w
. n-i.n—1 ,r1 ^ ^ u^+O)
^ = o j=o
(1 - uji t)L'
y
>
Interchanging the Q- and j-summations we see that the Q-summation gives zero for j ^ 0, leaving only the j = 0 term, which is (8). Eq. (8) can now be used to obtain recursion relations: Write N-l
(l + t + t2 + ... + tN~1)YJ
N-l
t°L'i PQL){s) = J2 t"L+^ PQL+1)(s).
Q=0
Q=0
280
(9)
Onsager's
Algebra and Partially
Orthogonal Polynomials
2133
Comparing powers of t, e.g. for L mod N = 0 this gives 1 1
/ 1 1 1
(i+i) :>(£+!)
S
I P^ \ (L) PI >w
(10)
s J \ p(L) j
V i
V^-W
\
For L mod N = k replace PQ —> PQ-k cyclically in both column vectors, keeping the same square matrix. Recursion relations not coupling Pk ' with different Q follow by the TV—fold application of these relations, leading to N + l-term recursion formulae. These can be transcribed into the corresponding formulae for the ITQ . For the Ising case Z2 these are of the Chebyshev type n(
L
+4)
4clXQ(i+2) + 4n!Q
0,
(11)
2 the ITQ form orthogonal sequences. However, for Z3 we have 11 and so for N (valid for all L > 0 and all Q):
n^ L+9) - 3(9c2 - 5)n^ L + 6 ) + 48 n^ L+3) - 64 n^L) = o
(12)
These n^ L) form 9 sequences, each labeled by (Lo, Q), where Q = 0,1,2 and L = 3J+LQ where LQ = 0,1,2, j = 0,1,2, The degrees of the polynomials appearing in this relation increase by two from the right to the left, but since the recursion is four-term, not three-term, these are not orthogonal sequences 21a . However, like (11) also (12) are of the most simple type b : all coefficients are independent of L. 4. Expansion in terms of Jacobi polynomials As the zeros of our IIQ are confined to and dense in the interval (—1,-1-1) we call this the basic interval like for orthogonal polynomials. Trying to determine (numerically) a weight function by the ansatz /_ 1 (1 + c ) a ( l — c)^ckU.Q (c) = 0 for k < bQtL fitting a and (3, we find (in the following we concentrate on the Z3-case) that there is an approximate solution very close to a = —/3 = | , but a and (3 come out to be slightly L and fc-dependent, in contrast to what is needed for orthogonality. However, for L —> oo and small fixed k, the solutions converge 1
(-
(L)
—-)
towards a = —f3 = 3 . So the n ^ seem to be close to Jacobi polynomials P^ 3 ' , but can we formulate an exact relation valid for finite L? Is there an exact property of the U.Q ' which replaces orthogonality? Numerical calculations 11 gave the surprising result that seemingly complicated HQ ' a If we consider Q = L mod 3, then we have only polynomials in c 2 , and the degrees 6jr,,Q are consecutive in powers of z = c 2 ("simple sets of polynomials") with integer coefficients. b For higher N we get similar N + l-term relations, e.g. for Z4: 4) +12) - 128(14c 2 - 1 7 ) n ^ + 8 ) - 2 0 4 8 c I I ^ + 4 ) + 4096 n ^ = 0. n (L+i6) _ 4 { 6 4 c 3 _ 5 6 c ) n < n ^
281
2134
G. von
Gehlen
Table 1.
Examples of Z3—polynomials rig '(c) and their Jacobi-components. Z3,
Q = L + 1 mod 3,
(a,/3) = ( § , - 1 )
n(c)
L
2-["/3in^(c)
3
9c + 3
[0,|]
4
27c2-18c-5
5
81 c 3 + 2 7 c 2 - 5 7 c - l l
[3,-f,fl [o--f-o,iJ
6
35c3+81c2-135c-21
[0,0,0,^]
7
3 6 c 4 - 2 - 3 5 c 3 - 540c 2 + 2 7 0 c + 43
fo l°!
8
3 7 c 5 + 3 6 c 4 - 2754 c 3 - 702 c 2 + 711 c + 85
9
3 8 c 5 + 3 7 c 4 - 7290 c 3 - 1782 c 2 + 1593 c + 171
[0,0,0,-2,0,^] [0u u0 0 - 5 1 0u 22^1
10
3 9 c 6 - 2 • 3 8 c 5 - 25515 c 4 + 14580 c 3
2 7 171 4 ' 14 '
729 7291 40 ' 70 J
L i i "i 40 1 > 56 J ro 2 7 135 243 9477 L°> 4 i 14 > 20 ' 385 ' 37 38 i 56 ' 3081
+7965 c 2 - 3 1 8 6 c - 3 4 1
can be written as a combination of just very few Jacobi polynomials, e.g. n(!12) = 3 n c 7 + 3 1 0 c 6 - 5 - 3 1 0 c 5 - 80919c4 +140697c 3 + 27459c2 -16839c - 1 3 6 5 = — ( l Pft~*] 728
. For polynomials TT(C) of degree n we use c :
- P^~^\ 7T(C) =
[TTo, TTi, . . . , 7Tn] = ^ 7 r
f e
P^.-^)
( c )
(13)
fc=0
and define a scalar product with Baxter's variable t = ((1 — c)/(l + c)) 1 / 3 (see (6)) as the weight function (here we will not need to specify the normalization):
WW -£'*(££
1/3 v r(l)r r((2~'(cj )f 7r ^(c)7T
0
- /— oo
6s Vx
;irW(c(s))irW(c(s)).
The second part of this definition shows that it makes sense also if we prefer to use polynomials in the variable s, and that it preserves the original Z3—symmetry. Since the Ilk ' satisfy the recursion relations (12) they can be written as determinants of band matrices with a bottom line specifying the initial conditions (which are the lowest L polynomials). Omitting the bottom line, we define j x j—band matrices and their determinants Rj = det | band( [64, 48, 3p, 1, 0], j)\ , e.g.
Re
3p 1 0 0 0 0 48 3p 1 0 0 0 64 48 3p 1 0 0 0 64 48 3p 1 0 0 0 64 48 3p 1 0 0 0 64 48 3p
where p = 9c — 5, so that the polynomials Rj depend only on c . Now we get the nine sequences of the IIQ ' as linear combinations of the Rj which satisfy appropriate c
For Jacobi series as a generalization of Taylor series, see Ch.7 of Carlson. 2 3 We use the standard normalization of Jacobi polynomials, see e.g. Rainville. 22 282
Onsager's Algebra and Partially Orthogonal Polynomials 2135 initial conditions. Abbreviating Qj = Rj + 8Rj-i l4 3,-) = 5 (Qj ~ 8 < 2 J - I + 16Qj-2);
these are found to be: n ( 3 j ) = 9 c ^ _ i ± 3Q,-_i; 2
n |3i+l) = ± 1 8 c ^ . _ 1 +
Q^. +
2 Q j l
.
n (3j+l) =
n l 3 j + 2 ) = 3c(Rj - 4R,-_i) ± (Q,- - 4Q j _i);
^. _
A Q j i
.
(M)
n^ 3 j + 2 ) = 3Q,-.
For j — 1,2 use i?o = 1; -ft-i = R-2 = 0. From (14) we see that only two of the 9 sequences are independent. There are relations like e.g. IJ^j+1'—U.i:'+1' = 2E4 . To get the Jacobi-expansion of the Ilk , we only need to expand Rj and cRj. Using Jacobi-components denned in (13), from explicit calculation (for j < 36), we find that for k < j (only there) the j-dependence of the (Rj)k obeys the simple rule: (Rj)k
= (-3) fc k\ (2k + 1) {(-8)1 ak + 4? rk} ;
1
2(-3)m
_
3llJUo(3n + l)
_
_
3 n ' = m ( 2 n + l)
2(-3)m~2 n*=m(2n)
It follows that the ak do not contribute to the A; < j-components of Qj, and, using the recursion relations for the Pk' \(Qj)k
= -(cRj)k
2
= \(c Rj+1)k
3
(c), we conclude that for k < j we have
= V(-3)k +
k\ (2k + l)rk.
(ny+\ = (n?\ = (ui* \ = (n^) i
It further follows that +2
=(n^ >) =o for k<j. k
i
k
All k < j components of II^3 , II2 3 and U^^2' are proportional to (Qj)kWe get zero overlap between a polynomial YLQ ' of degree bi,tQ with all polynomials ITQ, which have at least b/^Q vanishing low-fc components (these can only be polynomials which have &£/,Q' > 2bL:Q). One of many such relations is e.g. ( n g j ) I n g / ' } ) = 0 for Q = Q' = 1
and 2j < f - 1.
This property may be called "partial orthogonality". Further rules, this time valid for all k, regard the vanishing of many components for particular linear combinations: Defining Qf
= n<3j'> =9cRi-1+3Qj-1
Q{1] =
Qj +9cRj-!
- Qj-i
= U+\
a<+\ ..., a g ^ ] ;
= [oi-\
a{-\
...,
a{-] ],
we have checked up to j = 30 that ak ' = 0 for k < j and that all even (odd) k Jacobi components of Q+ (Q_ ) vanish. So we conjecture for all j , j '
(Q{i)\Q{P) = 0. One can check some special cases of these results in Table 1. It has been found numerically11 that a similar partial orthogonality appears also for the Zjv-Baxter polynomials with N = 4, 5, 6. For even values of N further relations emerge, but these will not be discussed here. 283
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G. von Gehlen
5. C o n c l u s i o n The polynomials which play a central role for t h e calculation of the energy eigenvalues of the superintegrable Z3—chiral P o t t s model are found to be related to Jacobi polynomials in a very peculiar way. M a n y integrals giving the Jacobi-coefficients of Baxter's polynomials a r e found t o vanish. T h e s e observations should have a deeper group-theoretical b a c k g r o u n d , but t h e underlying symmetry is not yet clear to us. By reducing the formulation of the problem t o some basic facts, the present analysis tries to prepare t h e g r o u n d for clarifying t h e s y m m e t r y involved.
Acknowledgements The author is grateful t o Prof. Mo-Lin Ge for t h e very friendly hospitality during the A P C T P 2001 N a n k a i Symposium in Tianjin. T h i s work has been supported by INTAS-OPEN-00-00055.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
F.Y. Wu and Y.K. Wang, J. Math. Phys. 17, 439 (1976) S. Ostlund, Phys. Rev. B24, 398 (1981) D.A. Huse, Phys. Rev. B24, 5180 (1981) D A . Huse and M.E. Fisher, Phys. Rev. B 2 9 , 239 (1984) P. Centen, M. Marcu and V. Rittenberg, Nucl. Phys. B205, 585 (1982) S. Howes, L.P. Kadanoff, and M. den Nijs, Nucl. Phys. B215 [FS7], 169 (1983) H. Au-Yang and J.H.H. Perk, Int. J. of Mod. Phys. B l l , 11 (1997) G. von Gehlen and V. Rittenberg, Nucl. Phys. B 2 5 7 [FS14], 351 (1985) L. Onsager, Phys. Rev. 65, 117 (1944) R.J. Baxter, Phys. Lett. 133A, 185 (1988). G. von Gehlen and S.-S. Roan, in "Integrable structures of exactly solvable twodimensional models etc.", NATO Science Series 7735, 155, Kluwer Acad. Publ.(2001) B. Davies, J. Phys. A: Math. Gen. 23, 2245 (1990) L. Dolan and G. Grady, Phys. Rev. D 2 5 , 1587 (1982) J.H.H. Perk, in Theta Functions Bowdoin 1987, Am. Math. Soc. (1989) E. Date and S.-S. Roan, J. Phys. A: Math. Gen. 33, 3275 (2000) H. Au-Yang, B.M. McCoy, J.H.H. Perk, S. Tang and M.L. Yan, Phys. Lett. 123A, 219 (1987) R.J. Baxter, J.H.H. Perk and H. Au-Yang, Phys. Lett. 128A, 138 (1988) G. Albertini, B.M. McCoy and J.H.H. Perk, (1989) Adv. Stud. Pure Math. 19, 1 (1989) S. Dasmahapatra, R. Kedem and B.M. McCoy, Nucl. Phys. B257 506 (1993) R.J. Baxter, J. Phys. A: Math. Gen. 27 1837 (1994) T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach (1978) E.D. Rainville, Special Functions, Chelsea Publ. Comp. (1971) B.C. Carlson, Special Functions of Applied Mathematics, Academic Press (1977)
284
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2137-2143 © World Scientific Publishing Company
T H E R M O D Y N A M I C S OF T W O C O M P O N E N T B O S O N S IN O N E D I M E N S I O N
SHI-JIAN GU Zhejiang Institute
of Modern Physics, Zhejiang University,
Zhejiang Institute
of Modern Physics, Zhejiang University,
Zhejiang Institute
of Modern Physics, Zhejiang University,
Zhejiang Institute
of Modern Physics, Zhejiang
Hangzhou
310027, P.R.
China
Hangzhou
310027, P.R.
China
Hangzhou
310027, P.R.
China
Hangzhou
310027, P.R.
China
YOU-QUAN LI
ZU-JIAN YING
XUE-AN ZHAO University,
Received 3 December 2001
On the basis of Bethe ansatz solution of two-component bosons with SU(2) symmetry and 5-function interaction in one dimension, we study t h e thermodynamics of the system at finite temperature by using the strategy of thermodynamic Bethe ansatz (TBA). It is shown that the ground state is an isospin "ferromagnetic" state by the method of TBA, and at high temperature the magnetic property is dominated by Curie's law. We obtain the exact result of specific heat and entropy in strong coupling limit which scales like T at low temperature. While in weak coupling limit, it is found there is still no Bose-Einstein Condensation (BEC) in such I D system.
1. Introduction A two-component Bose gas has been produced in magnetically trapped 87Rb by rotating the two hypernne states into each other with the help of slightly detuned Rabi oscillation field.1 It was noticed2 that the properties of such Bose system can be different from the traditional scalar Bose system once it acquires internal degree of freedom. Bethe ansatz solution of SU(2) two-component bosons in one dimension was obtained. 3 ' 4 It was pointed out that the ground state of such a system is an isospin "ferromagnetic" state 4 which differs from that of spin-1/2 fermions in one dimension, such as SU(2) Hubbard model, 5 etc. An interacting SU(2) boson field trapped in a one dimensional ring of length L 285
2138 S.-J. Gu et al. can be modeled by the following Hamiltonian H=
dx
(i)
/ •
a
a,b
where a,b = 1,2 denotes the ^-component of isospin. The Bethe ansatz equations (BAE) of eq. (1) are obtained as follows ikjL _ _ TT *j — h+ic -I-T kj — \y — ic/2 e ~ M kJ -h-ic 1 1fc,-- Xv + ic/2
l=\ ^~ki+
ic/2 1=1 A7 - \„ - ic
U
where M denotes the total number of down isospins. Eq. (2) differs from the BAE of scalar bosons with periodic condition. 6 The second equation of eqs. (2) of isospin rapidity A arises from the application of quantum inverse method, 7 which can be inferred from spin-1/2 fermions8 too. However, the symmetry of bosonic wave function gives the first term on the right side of first equation of eqs. (2), which does not appear in the BAE for fermions. 2. Thermodynamics at Finite Temperature The strategy we use here is the thermodynamic Bethe ansatz (TBA) which was pioneered by C. N. Yang and C. P. Yang for the case of the delta-function Bose gas. 1 0 It is used to derive a set of nonlinear integral equations called TBA equations, which describe the thermodynamics of the model at finite temperature. Moreover, the As can be complex roots which should form a "bound state" with other As11 when T ^ O , which arises from the consistency of both sides of the BAE. For ideal A strings of length m the rapidities are A™J = A£ + (n + 1 - 2j)iu + 0(exp(-SN)). Here u = c/2, a enumerates the strings of the same length m, and j = 1 , . . . ,m counts the As involved in the ath A string of the length m, A™ is the real part of the string. Taking logarithm of the BAE (2) by using string hypothesis we arrive at the following discrete Bethe ansatz equations
1-KIJ = kjL + 2^2e2(fc,- -ki)-J2Q"(ki I
2nJ: = 2 £ On(A£ - *,) - 2 £ l
where 0n(,x) = tan
1
(x/nu)
~ A")
an
AnltQt(Xna - Xlb)
bl,tytO
and
1, fori = n + 1,17i — /| Anit = { 2, for i = n + 1 - 2, • • •, \n - l\ + 2 0, otherwise. 286
(3)
Thermodynamics of Two Component Bosons
2139
Ij and J™ play the role of quantum numbers for charge rapidity and isospin rapidity respectively. In order to guarantee linear independence of wave function, all quantum number within a given set of {7} as well as that in {J} should be different. An arbitrary quantum number may be either in the set or not in the set. The former is called a root, the later is called a hole. In thermodynamic limit, the distribution of charge rapidities becomes dense and it is useful to introduce the density function for charge roots and holes respectively. We denote with p(k) and ph(k) the density function of charge roots and holes, in a similar way, with
=
(l/L)dI{k)/dk (l/L)dJn(X)/d\.
(4)
Then from eqs. (3) we obtain a set of coupled integral equations. p + ph=
±+K2(k)*p(k)-J2Kn(k)*an(k) n
h
on + o n= Kn(\) * p(X) - J2 AnitKt{\)
* ai(X)
(5)
where Kn(x) — nu/ir(n2u2 + x2), and * denotes the integral convolution. In terms of the density functions of charge and isospin roots, the kinetic energy per length has the form Ef./L = j k2p(k)dk, the total number of down isospins is M/L = ^2nn Jan(X)dX and the particle density of the model is D = N/L = J p(k)dk. If we consider the energy arising from the external field 0 which is the Rabi field in two-component BEC experiments, the internal energy of the model is E/L = f{k2 - Q)p(k)dk + J ]
2n0
/ andX.
(6)
And with the help of the approach first introduced by Yang and Yang, 10 the entropy of the present model at finite temperature is S/L = f[(p + ph) ln(p + / ) - plnp -ph \nph}dk
(7)
+ J2 /[(CT« + *£) l n K + o£) - ov, In(7„ - ohn In <%]d\. n
J
The Gibbs free energy of the model then is defined by F = E — TS — pN, where p, is the chemical potential. In order to obtain the thermal equilibrium, we minimize the free energy with respect to all the density functions subjects to the constraint (5). In addition, the total number of particles, the magnetization are kept to constant. For this purpose, the chemical potential p and external field O play the role of Lagrange multipliers. It is useful to define the energy potential for charge sector and isospin sector: K(jfe) = e e(fc ) /T = Vn(X)
=ahn{X)/an{X). 287
ph{k)/p{k) (8)
2140
S.-J. Gu et al.
Applying the minimum condition SF = 0 gives rise to a revised version of GaudinTakahashi equations TlnAv -
e(k) = k2-fi-Q-
TK2{k) * ln[l + KT 1 ]
-Tj2Kn(k)*H^-+V~1} n
lnr/i = —sech(7rA/2u) * ln[(l +
K_1)(1
+ %)]
ln77„= —sech(7rA/2u)*ln[(l+»fc,_i)(l + »M+i)]. 4u And these equations are completed by the asymptotic conditions lim [lnr]n/n} = 2x
(9)
(10)
where x — fi/T. Eqs. (9) can be solved by iteration. Note that eqs. (5) together with eqs. (9) completely determine the densities of charge roots and isospin roots in the state of thermal equilibrium. The Helmholtz free energy F = E — TS is given by F=
f ln[l + e-elT\dk.
nN-—
(11)
The above approach called TBA is universal for discussing the thermodynamics of one dimensional integrable model. Once eqs. (9) are solved, all thermodynamic quantities can be obtained from eq. (11) in principle. 3. Magnetic Property: Zero and High Temperature Limit: The state at zero temperature is the ground state. The Fermi surface is determined by e{kp) = 0. Since there is no hole under Fermi surface, we can take the energy potential K = ph/p as zero. As a result, from eqs. (9), it is easy to see r]n —> oo, and M = 0, the "ferromagnetic" ground state. The first equation of eqs. (9) becomes e0(k) = k2 - ix - il + K2(k) * e0(k)
(12)
which gives the solution of dressed energy, and the ground-state energy may be given in terms of €o
1
rhF
E0/L = — / e0(k)dk. (13) 27r J-kF Consequently, the ground state of ID SU(2) bosons is an isospin "ferromagnetic" state, which coincides with the analysis of Li et al. 4 Then the property of the model at T = 0 is the same as that of scalar bosons in one dimension which has been discussed extensively by Lieb and Liniger. 6 In the isospin space, however, the SU(2) symmetry of whole system around the ground state is broken. 288
Thermodynamics
of Two Component Bosons
2141
In the high temperature limit T —> oo (free isospins), however we can assume that all functions r]n(\) are independent of A. Then eqs. (9) can be written as follows,
r)l = (1 + m) nl=
(l + Tfe-xXl + Tfc+i)
(14)
where we have neglected the term (1 + K - 1 ) in the second equation of eqs. (9). The solution of j]n are then constants fixed by the field boundary condition (10) to be Vn =
'sinh(n + l)x]2 sinhx
1
After perform the Fourier transformation on eqs. (5), we get the solution of the densities of A n-strings,
"=
^sech[7rA/2u]*[^1+«7ti].
(16)
If we assume that an andCT£are independent of A or let c = 0, the total number of down isospins has the form,
£nan = ^ - ^ . „
m
e - ^
(17)
n
where nm is maximal length of A strings. In the absence of Rabi field, we have M/N = 1/2, the system at high temperature is a quasi "paramagnetic" state. If the external field 0 is small, expanding eq. (17) for small field x and integrating the equation over A space, we get the magnetization of the model. Let Mm be the total number of isospin rapidities in all n m -strings,
where the first term in the parentheses arises from self-magnetization, while the others are contributed by Rabi field. Eq. (18) indicates that the magnetic property of the model in high temperature regime dominated by Curie's law x oc 1/T. 4. Strong and Weak Coupling Limit: When 77 —>• 00, Kn(k) = 0, from eqs. (9) we have e = k2 - il - /i-
(19)
The free energy of the system (11) at low temperature now can be solved by integration by part, F,L
2 1,3/2
= ,D--y-^
289
2^2
r v
+ WJ-2
(20)
2142
S.-J. Gu et cd.
where the external field is set to zero. We can not deduce the specific heat directly from the free energy obtained above because the chemical potential is a function of temperature. Prom eqs. (5), the density of charge rapidity has the form -
1
1
Clearly, at zero temperature, the Fermi surface is just the square root of the chemical potential, so we have ^o = 7r 2 D 2 . At low temperature, however, it is determined by D = N/L = / p(k)dk. After integration, we have a temperature dependent chemical potential 2^2-, - 2
1
(22)
24/x2. J
Then the free energy becomes 2
7T
2j>2
Since by thermodynamics S = —OF/dT and Cv — TdS/dT, heat at low temperature is Fermi-liquid like S = CV = ^ .
we find the specific
(24)
It is the same as the result of one-component case, since for the strong coupling limit the isospin and the charge are decoupled completely, the contribution of isospin to the free energy vanishes. In order to discuss the possibility of the existence of BEC, we consider the problem in weak coupling limit u —> 0. And isospin-isospin reaches its maximal correlation. At low temperature, however, we do not take string hypothesis for simplicity. Because \im.c^oKn(x) = 5(x), together with eqs. (5) and eqs. (9), we obtain PW
_ 1 (3e»-l)(e-+l) '27r(3e^o+l)(l_e-eo)
^5)
where so = (k2 — n)/T. The positive definition of p(k) requires that the chemical potential is negative. As we known the density of scalar boson is 2np = 1/(1 — e~ e °) which prevents the BEC in ID and 2D system because of the infrared divergence. However, the density function (25) still does not resolve this problem. Consequently, BEC does not happen in this model yet. 5. Conclusion and Acknowledgment To summarize, we discussed the general thermodynamics of one dimensional SU(2) bosons with J-function interaction by using the strategy of TBA. It was shown that the ground state is an isospin "ferromagnetic" state which differs from the ground state of ID fermions, while at high temperature, it is "paramagnetic" state and 290
Thermodynamics of Two Component Bosons 2143 t h e magnetic property is d o m i n a t e d b y Curie's law. In s t r o n g coupling limit, we obtain the exact expression of t h e d e p e n d e n c e of chemical potential, entropy and specific heat on temperature which a r e Fermi-liquid like, while in weak coupling limit, we found the infrared divergence of charge roots density function prevents t h e existence of B E C . This work is supported by T r a n s - C e n t u r y Training P r o g r a m Foundation for the Talents and EYF98 of China Ministry of E d u c a t i o n . S J G t h a n k s D.Yang for helpful discussions.
References 1. J. E. Williams, M. J. Holland, Nature 4 0 1 , 568(1999); J. E. Williams, M. J. Holland, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358(1999). 2. T. L. Ho, Phys. Rev. Lett. 8 1 , 742(1998). 3. P. P. Kulish, Physica 3D 1, 246 (1981). 4. Y. Q. Li, S. J. Gu, Z. J. Ying and U. Eckern, submitted to P R B , [cond-mat/0107578]. 5. E. L. Lieb and F. Y. Wu, Phys. Rev. Lett 20, 1445 (1968). 6. E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605 (1963). 7. L. D. Faddev, in Recent Advances in Field Theory and Statistical Mechanics, ed. J.-B. Zuber and R. Stora (Elsevier, Amsterdam 1984) p. 569. 8. C. N. Yang, Phys. Rev. Lett. 19. 1312 (1967). 9. E. H. Lieb, Phys. Rev. 130, 1616 (1963). 10. C. N. Yang and C. P. Yang, J. Math. Phys. 10 1115 (1969). 11. M. Takahashi, Prog. Theor. Phys. 4 7 , 69 (1972).
291
Internationa] Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2145-2151 © World Scientific Publishing Company
R-MATRICES A N D T H E T E N S O R P R O D U C T G R A P H M E T H O D
MARK D. GOULD YAO-ZHONG ZHANG Centre for Mathematical Physics, Department of Mathematics, Brisbane, Qld 4072, Australia
University
of
Queensland,
Received 12 November 2001
A systematic method for constructing trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two affinizable representations of a quantum algebra or superalgebra has been developed by the Brisbane group and its collaborators. This method has been referred to as the Tensor Product Graph Method. Here we describe applications of this method to untwisted and twisted quantum affine superalgebras.
1. Introduction The (graded) Yang-Baxter equation (YBE) plays a central role in the theory of (supersymmetric) quantum integrable systems. Solutions to the YBE are usually called R-matrices. The knowledge of R-matrices has many physical applications. In one-dimensional lattice models, R-matrices yield the Hamiltonians of quantum spin chains.1 In statistical mechanics, R-matrices define the Boltzmann weights of exactly soluble models2 and in integrable quantum field theory they give rise to exact factorizable scattering S-matrices.3 So the construction of R-matrices is fundamental in the study of integrable systems. Mathematical structures underlying the YBE and therefore R-matrices and integrable models are quantum affine (super)algebras. A systematic method for the construction of trigonometric R-matrices arising from untwisted and twisted quantum affine (super)algebras has been developed in Refs. 4-9 (see also Ref. 10 for rational cases). This method is called the Tensor Product Graph (TPG) method. The method enables one to construct spectral dependent R-matrices corresponding to the (multiplicity-free) tensor product of any two affinizable representations of a quantum algebra or superalgebra. In this contribution, we describe the TPG method in the context of untwisted and twisted quantum affine superalgebras. Quantum superalgebras are interesting since the tensor product decomposition often has indecomposables and integrable models associated with them may in some instances be interpreted as describing strongly correlated fermion systems. 11 ' 12 293
2146
M. D. Gould & Y.-Z.
Zhang
2. Quantum Affine Superalgebras and Jimbo Equation Let us first of all recall some facts about the affine superalgebra G^k\ k — 1,2. Let Go be the fixed point subalgebra under the diagram automorphism of Q of order k. In the case of k = 1, we have Go = G- For k = 2 we may decompose G as Go © Gi, where [£i] C G\- Let tp be the highest root of Go = G for k = 1 and 0 be the highest weight of the ^-representation G\ for A; = 2. Quantum affine superalgebras Uq[G^] are g-deformations of the universal enveloping algebras U[Q^] of 5 ^ • We shall not give the defining relations for UglG^}, but mention that the action of the coproduct on its generators {hi, e^, fi, 0 < i < r} is given by A(/ii) = hi® 1 + 1® hi, A(ei) = ei®q^
+ q~^ ® eu
A(/i) = / j q^ + q~^ ® f{.
(1)
Define an automorphism Dz of Uq[G^] by £»z(e,) = zk5<°ei,
= z-kSi°fi,
Dz(fi)
Dz(hi) = K
(2)
Given any two minimal irreducible representations TT\ and 7r^ of Uq[Go] and their affinizations to irreducible representations of Uq[G^], we obtain a one-parameter family of representations A? of Uq[G^] on V(X) ® V(fj,) defined by A^(a)=7rx®7r^((I?z®l)A(a)),
Va € CM0(fc)],
(3)
x
where 2 is the spectral parameter. Let R ^(z) be the spectral dependent R-matrices associated with ir\ and 7rM, which satisfies the YBE. Moreover it obeys the intertwining properties:
i ? ^ ) A y a ) = (Ar)ya)i^(z) 13
(4)
XlJ,
which, according to Jimbo, uniquely determine R (z) up to a scalar function of z. We normalize Rxi*{z) such that Rx'1{z)R'iX{z-1) = I, where Rx»(z) = P Rxi*{z) with P : V(X) ® V(fj.) —> V(/J,) ® V(X) the usual graded permutation operator. In order for the equation (4) to hold for all a € Uq[G^] it is sufficient that it holds for all a G Uq(Lo) and in addition for the extra generator eo. The relation for eo reads explicitly Rx»(z)
(z7rx(eo)®Trli(qh°/2)+*x(q-h°/2)®^(eo))
= (^(e0)®irx(qh°'2)
+ znfi(q-h°/2)®7rx(e0))Rx»(z).
(5)
Eq.(5) is the Jimbo equation for Uq[Q^]. 3. Solutions to J i m b o Equation and Tensor Product Graph Method Let V(X) and V([i) denote any two minimal irreducible representations of Uq[G^\. Assume the tensor product module V(A) ® V(fi) is completely reducible into irre294
R-Matrices
and the Tensor Product Graph Method
2147
ducible Uq[Go\-mo&\Aes as V{\)®V{ii)
= ($V{v)
(6)
and there are no multiplicities in this decomposition. We denote by P^ the projection operator of V[X) ® V{y) onto V{v) and set P* M = R^(l) P^ = Ppx Rx»(l). We may thus write #"(*) = X > ( s ) P ^ ,
p„(l) = l.
(7)
Following our previous approach, 5 the coefficients pu(z) may be determined according to the recursion relation P {Z)
"
fWt
+
= zfM
e^zqW*
+ wfVV*""^
(8)
which holds for any v ^ v' for which P^
( ^ ( e o ) ® KMh°'2))
P?
± 0.
(9)
Here C(u) is the eigenvalue of the universal Casimir element of Go on V{v) and eu denotes the parity of V(u) C V"(A) V(JJ,). We note that eo <E> qrh°/2 transforms under the adjoint action of Uq[Go] as the lowest weight of So-niodule V(tp) [resp. V(6)} for A; = 1 (resp. k = 2) (i.e. as the lowest component of a tensor operator). Throughout we adopt the notation <<*>±=
l±zqa •
0
,
(10)
so that the relation (8) may be expressed as
^./m^m)
MJ,
(11)
To graphically encode the recursion relations between different pu we introduce the Extended TPG for Uq[G(1)] and Extended T w i s t e d T P G for Uq[G{2)}. Definition 1: The Extended T P G associated to the tensor product V(X) V(fi) is a graph whose vertices are the irreducible modules V(v) appearing in the decomposition (6) of V(X) <8> V{p). There is an edge between two vertices V(y) and V{v>) iff V(v')^Vadj®V{v)
and
e(i/)e(«/) = - 1 .
(12)
The condition (12) is a necessary condition for (9) corresponding to Uq[G^] to hold. Definition 2: The Extended Twisted T P G which has the same set of nodes as the twisted TPG but has an edge between two vertices v ^ v' whenever V(i/) C V(6) V(i/) 295
(13)
2148
M. D. Gould & Y.-Z.
Zhang
and _ J + 1 if V(v) and V(v') are in the same irreducible representation of Q \ — 1 if V(v) and V(i/) are in different irreducible representations of Q. (14) The conditions (13) and (14) are necessary conditions for (9) corresponding to Ug[gW] to hold. We will impose a relation (8) for every edge in the extended (twisted) TPG but we will be imposing too many relations in general. These relations may be inconsistent and we are therefore not guaranteed a solution. If however a solution to the recursion relations exists, then it must give the unique correct solution to the Jimbo's equation. 4. Examples of R-matrices for
Uq[gl(m\n)^]
Throughout we introduce {ej}^ =1 and {£j}? = i which satisfy (ej,ej) = 6ij, (6i,Sj) = —Sij and (u,Sj) = 0. As is well known, every irreducible representation of Uq[gl(m\n)] provides also an irreducible representation for Uq[gl(m\n)^]. Here, as examples, we will construct the R-matrices corresponding to the following tensor product: rank a antisymmetric tensor with rank b antisymmetric tensor of the same type. Without loss of generality, we assume m>a>b and the antisymmetric tensors to be contravariant. The tensor product decomposition is V(\a)®V(\b)
= ($V(Ac)
(15)
c
where, when a + b < m, b
a+c
Ab = ^ € i ,
b—c
Ac = ^ e i + ^ e i ,
i=l
i=l
c = 0,l,---,6
(16)
j=l
and when a + b > m, a+c
b—c
Ac=^ei + ^ei, t=i m
c = 0,1, • ••,m-
Ac = ^ C J + ^ e j + (a + c - m ) 5 i , t=i
a
»=i b—c
c — m-a
+ l,---,b
(17)
»=i
The corresponding TPG is V(\a)®V(\b)
(18) A0 Ai A 6 _i Afc which is consistent; such is always the case when a graph is a tree (i.e. contains no closed loops). From the graph we obtain b
#«•**(x) = ^2Y[{2i + a-b)_ pfcXh c=0 i = l 296
(19)
R-Matrices and the Tensor Product Graph Method 2149
The a = b = 1 case had been worked out before, which is known to give rise to the Perk-Schultz model R-matrices. 14,15 5. Examples of R-matrices for C/9[gZ(n|n)^2)] To begin with, we introduce the concept of minimal representations. By minimal irreducible representations of Q, we mean those irreducible representations which are also irreducible under the fixed subalgebra QQ. We can determine R-matrices for any tensor product V(\a)®V{\b) of two minimal representations V(Xa), V(Xb) of Uq[gl(m\n)^], where V(Xa) is also irreducible model under Uq[osp(m\n)} with the corresponding Uq[osp(m\n)\ highest weight Aa = (6|a, 6). Recall that for our case QQ = osp(m\n) and 6 = 8i + 82. Below we shall illustrate the method for the interesting case of a — b, m = n > 2, where an indecomposable appears in the tensor product decomposition. The decomposition of the tensor product of two minimal irreducible representations of Uq[osp(m\n)]:9 a
V(Xa)®V(Xb)
c
= Q)(£)V(k,a
+ b-2cy,
(20)
c=0 k=0
here and throughout V(a, b) denotes an irreducible Uq[osp(m\n)} module with highest weight Xat(, = (0|a + 6, a, 0). Note that one can only get an indecomposable in (20) when m = n > 2 and a + b - 2c = 0. Since a < b, c < a, this can only occur when a = b and c = a. In that case the C/q[osp(m|n)]-modules V(k,0), k — 0,1, will form an indecomposable. From now on we denote by V this indecomposable module, and write the Uq[osp(n\n)} module decomposition (20) as V{\a)®V{\a) = @V(v)®V,
(21)
where the sum on v is over the irreducible highest weights. Note that V contains a unique submodule V^#i + ^2) which is maximal, indecomposable and cyclically generated by a maximal vector of weight 81+82 such that V/V{8\+52) = V(0j0) (the trivial f/q[osp(n|n)]-module). Moreover V contains a unique irreducible submodule V(0|0) C V(8i +82)- The usual form of Schur's lemma applies to V{8\ +82) and so the space of Uq[osp(n\n)] invariants in End(V) has dimension 2. It is spanned by the identity operator I together with an invariant ./V (unique up to scalar multiples) satisfying NV = V{0\6)cV(81
+ 82),
NV(81 + 82) = (0).
(22)
It follows that TV is nilpotent, i.e. TV2 = 0. We can show9 that the minimal irreducible Uq[osp(n\n)} modules, V(Xa), with highest weight Aa, are affinizable to carry irreducible representations of Uq[gl(n\n)(2'] We now determine the extended twisted TPG for the decomposition given by (21). 297
2150 M. D. Gould & Y.-Z. Zhang
We note that V can only be connected to two nodes corresponding to highest weights (opposite parity), + 62) (same parity),
(c, k) = (a - 1,0) (c, k) — (a, 2).
,
We thus arrive at the extended twisted T P G for (21), given by Figure 1. k =0 c= 0
c=l
c= 2
k =a—1 c= a— 1 A; = a c= a Fig. 1. The extended twisted TPG for Ug[gl(n\n)^] (n > 2) for the tensor product V(Xa) ® y(A a ). The vertex labelled by the pair (c, k) corresponds to the irreducible Uq[osp(n\n)] module V(k,2a — 2c) except for the vertex corresponding to c = a, k = 1, which has been circled to indicate that it is an indecomposable [79[osp(n|n)]-module. It can be shown that the extended twisted TPG is consistent, i.e. that the recursion relations (8) give the same result independent of the path along which one recurs. To prove this it suffices to show for each closed loop of four vertices in the graph, that the difference in Casimir eigenvalues for osp{n\n) along one edge equals the difference along the opposite edge. Let Py = PyaXa be the projection operator from V(A n )
+ Pv{z)Pv 298
+ J2 P»(z)p»-
(24)
R-Matrices and the Tensor Product Graph Method
2151
T h e coefficients pv{z) can b e obtained recursively from the extended twisted T P G . However, t h e coefficients PN{Z) a n d pv{z) cannot be read off from t h e extended twisted T P G since the corresponding vertex refers to an indecomposable module. R a t h e r t h e y are determined by t h e a p p r o a c h 1 6 to Uq[gl(2\2)^]. The result is 9 a ' c ' c—fe
R{z) = pN{z)N
+ pv(z)Pv
+ E E I 1 ^ -
2a
>+
c=0 fe=0 j = l c
Yl(i-2a-l)-P{2a-2c+k)s1+kS2,
(25)
i=l
where t h e p r i m e s in the sums signify t h a t t e r m s corresponding to c = a w i t h k = 0,1 are o m i t t e d from t h e sums, a n d pv(z),
PN(Z) are given by
2 a—1 q
PN{z)
= (-l)V
a—1
j=i a 2
t=i
\ ^ z Pv{z). 1+2
(26)
Acknowledgements This work h a s b e e n financially s u p p o r t e d by the Australian Research Council.
References 1. E. K. Sklyanin, L. A. Takhtadzhyan and L. D. Faddeev, Theor. Math. Phys. 40, 194 (1979). 2. R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982). 3. A. B Zamolodchikov, Al. B. Zamolodchikov, Ann. Phys. 120 (1979) 253. 4. R. B. Zhang, M. D. Gould and A. J. Bracken, Nucl. Phys. B354, 625 (1991). 5. G. W. Delius, M. D. Gould and Y.-Z. Zhang, Nucl. Phys. B432, 377 (1994). 6. G. W. Delius, M. D. Gould, J. R. Links and Y.-Z. Zhang, Int. J. Mod. Phys. A10, 3259 (1995). 7. G. W. Delius, M. D. Gould and Y.-Z. Zhang, Int. J. Mod. Phys. A l l , 3415 (1996). 8. G.M. Gandenberger, N. J. MacKay and G. M. T. Watts, Nucl. Phys. B 4 6 5 , 329 (1996). 9. M. D. Gould and Y.-Z. Zhang, Nucl. Phys. B566, 529 (2000). 10. N. J. MacKay, J. Phys. A 2 4 , 4017 (1991). 11. F. H. L. Essler, V. E. Korepin and K. Schoutens, Phys. Rev. Lett. 68, 2960 (1992). 12. A. J. Bracken, M. D. Gould, J. R. Links and Y.-Z. Zhang, Phys. Rev. Lett. 74, 2768 (1995). 13. M. Jimbo, Int. J. Mod. Phys. A 4 , 3759 (1989). 14. J. H. H. Perk and C. L. Schultz, Phys. Lett. A84, 407 (1981). 15. V. V. Bazhanov and A. G. Shadrikov, Theor. Math. Phys. 73, 1302 (1988). 16. M. D. Gould, J. R. Links, I. Tsohantjis and Y.-Z. Zhang, J. Phys. A30, 4313 (1997).
299
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 2153-2159 © World Scientific Publishing Company
FREE FIELD A N D P A R A F E R M I O N I C REALIZATIONS OF TWISTED su(3)£. 2) C U R R E N T A L G E B R A
XIANG-MAO DING * M A R K D. GOULD YAO-ZHONG ZHANG Centre for Mathematical Physics, Department of Mathematics, Brisbane, Qld 4072, Australia
University
of
Queensland,
Received 13 November 2001 Free field and twisted parafermionic representations of twisted su(3), current algebra are obtained. The corresponding twisted Sugawara energy-momentum tensor is given in terms of three (/3,7) pairs and two scalar fields and also in terms of twisted parafermionic currents and one scalar field. Two screening currents of the first kind are presented in terms of the free fields.
1. Introduction Infinite dimensional algebras, such as Virasoro algebra and affine algebras are algebraic structures in conformal field theories (CFT) in two dimensional spacetime. 1-3 They also play a central role in the study of string theory. 4 It is well-known the untwisted current algebra can be realized by at least two different ways: one is the free field representation, 5 and the other is the parafermion representation. 6 The free field realization is a common approach used in conformal field theories. 5 The free field representations for untwisted affine algebra have been extensively studied. The simplest untwisted case su(2)^ was first treated in Ref. 5, and the generalization to su(n)^ was given in Refs. 7-17. The Zk parafermions are generalizations of the Majorana fermions, and the Z\. parafermion models are extensions of the Ising model, which corresponds to the case fc = 2. 1 9 , 2 0 Parafermions are related to the exclusion statistics introduced by Haldane. 21 The recently study shows that twisted affine algebras are useful in the description of the entropy of Adsz black hole. 18 However, little is known for free field and parafermionic representations of twisted affine algebras. So it is an interesting problem to investigate such realizations. In this contribution we consider the simplest twisted affine algebra sw(3)^.'. We construct two different realizations for this algebra: one is free field representation, 'Permanent address: Institute of Applied Mathematics, Academy of Mathematics and System Sciences; Chinese Academy of Sciences, P.O.Box 2734, 100080, China. 301
2154
X.-M. Ding, M. D. Gould & Y.-Z. Zhang
which is another version of one given in Ref. 22, and another is twisted parafermionic representation by using the twisted parafermionic currents proposed in Ref. 23. Moreover we give the screening currents in terms of the free fields. 2. Twisted current sw(3)^ ' algebra We consider the simplest twisted affine Lie algebra
(2)
SM(3))J.
. We decompose su(3)
as su(3) = go © ffi
(1)
where 50 = su(2) is the fixed point subalgebra under the automorphism and #1 is the five dimensional representation oi go; go and g\ satisfy [gi,gj] C g(i+j) mod 2- Using the notation in Ref. 22, we chose e, / and h to be the bases in go and e, / , E, F and h the bases of g\. Then the commutators of su(3)k can be expressed as >X,zn®Y]
= z m+n
,
(X\Y) [X,Y] + 2kmSm+n,o-
(2)
Where m £ Z if X € go, and m € Z + \ if X € gi. Denote the currents corresponding to e, h, f by j+(z), j°(z), j~(z), and to e, h, / , respectively. Then (2) can be written in terms of the following OPE's: j+(z)j
Ak (z — w)2
(w)
+
1 r j » + -., (z — w)"
Ak
J+(z)J-(w) J++(z)J-(w)
J°(z)J±(w)
1
=
^ + T-±-J°(w) + ..., (z — w)z (z — w) = T-^L- 2 + j-^—fiw) + ..., (z — w) (z — w) 8k ±2 • j ± H + ..., j°(z)j°(w) (z-w] (z — w)2 ±6 ±6 ± = J±(w) j H + ... (z — w) (z — w)
J+(z)J—(w)
= 7-l—j-(w) (z-w)' =
j+{z)J-(v,)
= j ^ J ° W
j+{z)J—{w)
= j-=*—J-(w) +. (z — w) ±2 = J±(w) + ... (z — w) 2Ak = r + .... (z — w)
fiz^iw) J°{z)J°{w)
(z — w)
J++(w)
J-{z)J++{
+.
j+{z)J+{w)
j+(w)+ ..... (z — w) -2 j (z)J (w) = J—{w) + ..., (z — w)
+. + ...
+ ...,
J+(z)j-(w)
=
j-(z)J++{w) fiz^iw)
302
(z — w) =
=
J°(w) +. — J+(w)
±4 j (z — w)
± ±
H + ...:
Free Field and Parafermionic
Realizations
2155
All other OPE's contain trivial regular terms only. Here and throughout "..." stands for regular terms. 3. Wakimoto free field realization of the twisted affine currents To obtain a free field realization of the twisted su(3yk' currents, we follow the procedure adopted in Ref. 22 but begin with a different Fock space. The Fock space is constructed by the repeated actions of / , / , F on the highest weight state v\ with highest weight A. The highest weight states are determined by evA = evA = EvA = 0,
hvA = (A, ai)v A ,
hvA = (A, a2)vA
(3)
where a\ and a2 are roots associated with h and h. Set \n, m, I > = fnFmflVAThis choice is different from the one used in Ref. 22. As we shall see, this choice gives another free field realization of the twisted current algebra. Introduce three /3j pairs and two scalar fields (j>a, a = 1, 2. (/%; ji) pairs have conformal dimension (1; 0). ft(2)7; H = - T i W f t H = - - ^ - , *, j = 0,1,2 z—w 4>a{z)4>b{w) = -25a,b ln(z - w), a, b = 0,1
(4)
Introduce the notation ei = | ( 1 , 1 ) ; e2 = - ^ ( 1 , - 1 ) ; and l> = (4>o,4>±). Then we have e\ • $(z)ei •
e2 • §>{z)e2 • $(w) = - 3 ln(z — w); e*i • $(z)e2 • $>(w) = 0.
We find the Wakimoto free field realization of su(3)jj.' in terms of the eight free fields: j+(z)=
A>(z)+2ft(z)7i(*),
j°(z) = 2p0(z)l0(z) j~(z)
+ 2(31{z)ll{z)+Ap2{z)l2{z)
+ — (ei • a+
id$(z)),
= -Po(z) (7o2W + 3 7 l 2 W) - 2fa{z) (370(2)71 W - 72(2)) -2/32(z) (37g(2)7i W + 7?(*)) - 4(fc + l)57o(«) (ei • id$(z))nf0(z)
j\z)
(e2 • id$(z))7i(;z),
= 6/3o(2)7iW + 6/?i(2)70(2) + 6ft(z) (7o2(2) + 7i2(2))
+ — (e2-id$(z)); a+ J+(Z) = /3l(2),
(5)
J - ( z ) = -2A)(z) (70(2)71(2) +72(2)) - A(2) (37o2(2) +7i 2 (2)) -4ft(z) (7o3(2) +71(2)72(2)) -4fc9 7 i(2), (ei • ia$(z))7!(2;) 303
(e2 •
id$(z))j0(z);
2156 X.-M. Ding, M. D. Gould & Y.-Z. Zhang
J—(z)
= 20o(z) ( 7 g(*)7i(*) +7i 3 (z) -
27o(zh2(z))
2
- 2 f t (z) (
= T^zftz-"-1;
Ja(z) = ZnzZ+1/2JZz-n-\
(6)
4. Twisted parafermions and parafermionic realization In this section we use the twisted parafermionic currents proposed in Ref. 23 to give another realization of the twisted sw(3)J.' currents. First, we recall the twisted parafermion current algebra given in Ref. 23 reads,
(z — wy
z—w
_„A"72fc ==fJML Wt>i.H(z-")""* r wz ^ + P H where I, I' — ± 1 and I, I' — 6, ± 1 , ±2; sij', ej p and et p are structure constants. If we denote ipi or ip^ by V&a, then we can rewrite the above relations as: 00
- w)ab'2k
*a(z)Vb(w)(z
=
£ (* - ™)n[*a^\-n, n=-2
(8)
For consistency, ea,b must have the properties: ea,b = Sb,a = —£-a,-& = £-a,a+6 and ea ~a = 0. As a CFT, we can set the structure constants as e
l,-2
— e
l,l —e-l,2
_
1
—
\
rri
3 -1,-1 - c o , I - - y ^fe-
(9)
Then we may obtain a representation of the twisted su(3)k current algebra with the help of the twisted parafermionic currents. The result is j+(z) = 2Vktl>i(z)e^Mz),
J~(z) = 2 v / f c ^ _ 1 ( z ) e - ^ ' W z ) , 304
Free Field and Parafermionic Realizations 2157 f(z)
= 2V2Jfet0fo(z), = 2Vki>_i(z)e-^Mz\
J-(z) J—(z)
2Vk^i(z)e^Mz),
J+(z) = J++(z)
= 2y/kip_i(z)e-i^^z\
2Vk^(z)ei^Mz),
=
J°(z) = 2v/6fcVo(2)-
where o is an U(l) current obeying (J)Q{Z)(J)Q{W) = — ln(z — w). It can be checked that the above currents satisfy the OPEs of the twisted su(3)j.' currents algebra.
5. Twisted stress energy tensor It is well known that Virasoro algebras are related to currents algebras via the so called Sugawara construction. In the present case, the twisted Sugawara construction of the energy-momentum tensor is given by T(z)
X
\f{z)f{z)
8(fc + 3)
+ \j°{z)J\z)
+2J-(z)J+(z)
+
2j-(z)j+(z)
+ 2J—(z)J++(z)}
:,
(10)
where : : implies the normal ordering. The above expression can be rephrased through the ffy pairs and the scalar field 3>. We obtain T(z) = - : \p0(z)d7o(z)
+ (31(z)d11(z)
+ - : (Si • id$(z))
+ f32(z)dl2(z)j
: + - : (e 2 • id$(z))
:
:
- 4 a + (gi • id2$(z)V
(11)
On the other hand, the energy-momentum tensor in the twisted parafermionic realization is given by T(z) = T+ -
: d
(12)
where 2fc + 6
( 13 )
E[*-*-Jo
is the energy-momentum tensor of the twisted parafermion currents obtained in Ref. 23. Following the standard practice, we get the OPE of the energy-momentum tensor,
n,m«) =
^
+
*n=L
{z — wp
+
[z — wY
«W + ..., z —w
where c = 8k/(k + 3) is the central charge for the Virasoro algebra. 305
(14)
2158
X.-M.
Ding, M. D. Gould & Y.-Z.
Zhang
6. T w i s t e d screening currents An important object in the free field approach is the screening current. Screening currents are a primary fields with conformal dimension 1, and their integration give the screening charges. They commute with the affine currents up to total derivatives. These properties ensure that screening charges may be inserted into correlators while the conformal or affine Ward identities remain intact. For the present case, we find the following screening currents S±(z) = : [2fo(z)7o(z)+Pi(z)±l3o(z)]S±(z)
:,
(15)
where S±(z) = e - ^ + ^ i - ^ W i ^ W ) .
(16)
The O P E of the twisted screening currents with the twisted affine currents are T(z)S±(w)
= dw
(-^—S±(w)
j+(z)S±(w)
=
...,
j°(z)S±{w)
=
...,
j-(z)S±(w)
= dw (±^L--L-S±(w)^
J++(z)S±(w)
=
...,
J (z)S±(w)
=
...,
J°(z)S±(w)
=
...,
J-(z)S±(w)
= dw
+
J—(z)S±(w)
^ _ 1 _ 5
= dw (4"—^\ Oil Z
±
H )
+ ...,
+...,
(17)
[ 7 o H T 7i N ] S±(w)) IV
+...; J
The screening currents obtained here are the twisted versions of the first kind screening currents. 7 Acknowledgements This work is financially supported by Australian Research Council. One of the authors (Ding) is also supported partly by Natural Science Foundation of China, and a Foundation from AMSS. References 1. A. A. Belavin. A. M. Polyakov, A. B. Zamolodchikov, Nucl. Phys. B241, 333 (1984). 2. V. G. Kac, Infinite Dimensional Lie Algebras, third ed., (Cambridge University press, Cambridge 1990). 3. Ph. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, (Springer, 1997). 306
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21. 22. 23.
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